What Is The Sum Of The Terms Of The Series (Use The Formula): 7 11 15 . . . 35

Arithmetic Sequences

An arithmetic sequenceA sequence of numbers where each successive number is the sum of the previous number and some constant d. , or arithmetic progressionUsed when referring to an arithmetics sequence. , is a sequence of numbers where each successive number is the sum of the previous number and some constant d.

$$a_{\mathrm{due\; north}}=a_{\mathrm{northward}-1}+dArithmetic\mathrm{South}\mathrm{due\; east}que\mathrm{due\; north}c\mathrm{east}$$

And considering $a_{\mathrm{northward}}-a_{\mathrm{northward}-1}=d$ , the constant d is called the common differenceThe constant d that is obtained from subtracting any two successive terms of an arithmetic sequence; $a_n-a_{\mathrm{due\; north}-1}=d.$ . For example, the sequence of positive odd integers is an arithmetics sequence,

$$1,3,5,\mathrm{vii},\mathrm{nine},\dots$$

Here $a_{\mathrm{i}}=1$ and the difference between any two successive terms is two. We can construct the general term $a_{\mathrm{north}}=a_{\mathrm{due\; north}-1}+2$ where,

$$\begin{array}{ccc}\hfill a_1& =& 1\hfill \\ \hfill a_{\mathrm{two}}& =& a_1+\mathrm{two}=1+\mathrm{two}=\mathrm{three}\hfill \\ \hfill a_3& =& a_2+2=3+\mathrm{two}=5\hfill \\ \hfill a_4& =& a_3+\mathrm{two}=5+2=7\hfill \\ \hfill a_{\mathrm{five}}& =& a_{\mathrm{iv}}+2=7+2=\mathrm{ix}\hfill \\ & ⋮& \hfill \end{array}$$

In general, given the first term $a_{\mathrm{i}}$ of an arithmetics sequence and its common difference d, we tin write the following:

$$\begin{array}{ccc}\hfill a_{\mathrm{ii}}& =& a_{\mathrm{ane}}+d\hfill \\ \hfill a_3& =& a_2+d=\left(a_1+d\right)+d=a_{\mathrm{i}}+\mathrm{ii}d\hfill \\ \hfill a_{\mathrm{iv}}& =& a_3+d=\left(a_1+2d\right)+d=a_1+3d\hfill \\ \hfill a_5& =& a_{\mathrm{four}}+d=\left(a_1+\mathrm{three}d\right)+d=a_1+4d\hfill \\ & ⋮& \hfill \end{array}$$

From this we see that whatsoever arithmetics sequence tin can be written in terms of its first element, mutual difference, and index equally follows:

$$a_{\mathrm{due\; north}}=a_{\mathrm{i}}+\left(\mathrm{north}-\mathrm{one}\right)dArithm\mathrm{east}ticS\mathrm{eastward}quence$$

In fact, any general term that is linear in northward defines an arithmetic sequence.

Example 1

Discover an equation for the full general term of the given arithmetic sequence and use it to calculate its 100^th term: $\mathrm{seven},10,13,16,19,\dots$

Solution:

Begin by finding the mutual divergence,

$$d=\mathrm{x}-\mathrm{seven}=3$$

Note that the departure between any two successive terms is 3. The sequence is indeed an arithmetic progression where $a_{\mathrm{ane}}=\mathrm{vii}$ and $d=\mathrm{iii}.$

$$\begin{array}{ccc}\hfill a_n& =& a_{\mathrm{one}}+\left(\mathrm{due\; north}-\mathrm{ane}\right)d\hfill \\ & =& \mathrm{seven}+\left(n-1\right)\cdot 3\hfill \\ & =& 7+3\mathrm{north}-\mathrm{three}\hfill \\ & =& \mathrm{iii}n+4\hfill \end{array}$$

Therefore, we tin write the general term $a_n=3n+\mathrm{four}.$ Have a minute to verify that this equation describes the given sequence. Utilise this equation to notice the 100^th term:

$$a_{100}=3\left(100\right)+4=304$$

Answer: $a_n=3n+\mathrm{iv}$ ; $a_{100}=304$

The common difference of an arithmetic sequence may be negative.

Case ii

Discover an equation for the general term of the given arithmetic sequence and use information technology to calculate its 75^thursday term: $\mathrm{half-dozen},4,\mathrm{ii},0,-\mathrm{two},\dots$

Solution:

Brainstorm past finding the common difference,

$$d=4-6=-\mathrm{two}$$

Next find the formula for the general term, here $a_{\mathrm{i}}=\mathrm{half-dozen}$ and $d=-\mathrm{ii}.$

$$\begin{array}{ccc}\hfill a_n& =& a_1+\left(n-\mathrm{one}\right)d\hfill \\ & =& \mathrm{six}+\left(n-1\right)\cdot \left(-2\right)\hfill \\ & =& 6-2n+2\hfill \\ & =& 8-\mathrm{ii}n\hfill \end{array}$$

Therefore, $a_n=8-2\mathrm{due\; north}$ and the 75^th term can exist calculated as follows:

$$\begin{array}{ccc}\hfill a_{75}& =& 8-2\left(75\right)\hfill \\ & =& \mathrm{viii}-150\hfill \\ & =& -142\hfill \end{array}$$

Respond: $a_n=8-2n$ ; $a_{100}=-142$

The terms between given terms of an arithmetic sequence are called arithmetic waysThe terms between given terms of an arithmetics sequence. .

Example 3

Find all terms in between $a_1=-8$ and $a_{\mathrm{vii}}=10$ of an arithmetic sequence. In other words, find all arithmetic means between the ane^st and 7^th terms.

Solution:

Begin by finding the common difference d. In this case, we are given the first and seventh term:

$$\begin{array}{cccc}\hfill a_n& =& a_{\mathrm{i}}+\left(\mathrm{due\; north}-\mathrm{one}\right)d\hfill & Usen=7.\\ & & & \\ \hfill a_{\mathrm{vii}}& =& a_1+\left(\mathrm{seven}-1\right)d\hfill & \\ \hfill a_{\mathrm{seven}}& =& a_1+\mathrm{vi}d\hfill & \end{array}$$

Substitute $a_{\mathrm{one}}=-8$ and $a_7=10$ into the to a higher place equation and so solve for the mutual difference d.

$$\begin{array}{ccc}\hfill 10& =& -\mathrm{eight}+\mathrm{vi}d\hfill \\ \hfill \mathrm{xviii}& =& 6d\hfill \\ \hfill 3& =& d\hfill \end{array}$$

Side by side, apply the outset term $a_{\mathrm{ane}}=-8$ and the common departure $d=\mathrm{iii}$ to discover an equation for the nth term of the sequence.

$$\begin{array}{ccc}\hfill a_{\mathrm{due\; north}}& =& -8+\left(n-\mathrm{ane}\right)\cdot 3\hfill \\ & =& -8+\mathrm{iii}\mathrm{due\; north}-3\hfill \\ & =& -11+3\mathrm{due\; north}\hfill \end{array}$$

With $a_n=3n-\mathrm{eleven}$ , where n is a positive integer, find the missing terms.

$$\begin{array}{c}{a}_1=3\left(1\right)-11=\mathrm{three}-11=-8\\ \begin{array}{c}{a}_{\mathrm{ii}}=3\left(2\right)-11=\mathrm{vi}-11=-5\\ a_3=3\left(\mathrm{three}\right)-\mathrm{eleven}=\mathrm{ix}-\mathrm{eleven}=-2\\ a_{\mathrm{iv}}=\mathrm{iii}\left(\mathrm{iv}\right)-11=12-11=1\\ a_{\mathrm{five}}=3\left(5\right)-\mathrm{eleven}=15-11=4\\ a_6=3\left(6\right)-11=18-11=7\end{array}\}arith\mathrm{thousand}etic\mathrm{one\; thousand}\mathrm{due\; east}ans\\ a_7=\mathrm{three}\left(7\right)-11=21-11=10\end{array}$$

Respond: −5, −2, 1, 4, 7

In some cases, the first term of an arithmetic sequence may not exist given.

Example four

Discover the general term of an arithmetics sequence where $a_3=-\mathrm{i}$ and $a_{\mathrm{x}}=48.$

Solution:

To make up one's mind a formula for the general term we need $a_{\mathrm{ane}}$ and $d.$ A linear arrangement with these as variables can be formed using the given data and $a_n=a_1+\left(n-1\right)d$ :

$$\begin{array}{cccc}\{\begin{array}{c}{a}_{\mathrm{iii}}=a_1+\left(3-1\right)d\\ a_{10}=a_1+\left(\mathrm{ten}-1\right)d\end{array}& \underset{}{\Rightarrow }& \{\begin{array}{c}-\mathrm{ane}=a_1+2d\\ 48=a_{\mathrm{i}}+9d\end{array}& \begin{array}{c}U\mathrm{southward}e{a}_3=-\mathrm{ane}.\\ U\mathrm{southward}e{a}_{10}=48.\end{array}\end{array}$$

Eliminate $a_{\mathrm{one}}$ by multiplying the first equation by −1 and add the result to the 2d equation.

$$\begin{array}{l}\{\begin{array}{ccc}-1& =& a_1+\mathrm{two}d\\ 48& =& a_{\mathrm{i}}+9d\end{array}\begin{array}{c}\underset{}{\stackrel{\times \left(-\mathrm{ane}\right)}{\Rightarrow }}\\ \end{array}\underset{¯}{\begin{array}{c}\\ +\end{array}\{\begin{array}{ccc}1& =& -a_{\mathrm{ane}}-2d\\ 48& =& a_1+9d\end{array}}\\ \begin{array}{ccc}49& =& 7d\\ 7& =& d\end{array}\\ \end{array}$$

Substitute $d=\mathrm{seven}$ into $-\mathrm{i}=a_{\mathrm{one}}+\mathrm{two}d$ to find $a_1.$

$$\begin{array}{ccc}\hfill -\mathrm{one}& =& a_{\mathrm{ane}}+2\left(7\right)\hfill \\ \hfill -1& =& a_1+14\hfill \\ \hfill -15& =& a_{\mathrm{ane}}\hfill \end{array}$$

Adjacent, use the first term $a_1=-15$ and the common difference $d=\mathrm{seven}$ to find a formula for the general term.

$$\begin{array}{ccc}\hfill a_{\mathrm{northward}}& =& a_1+\left(\mathrm{north}-1\right)d\hfill \\ & =& -15+\left(\mathrm{northward}-1\right)\cdot 7\hfill \\ & =& -\mathrm{xv}+7n-7\hfill \\ & =& -22+7n\hfill \end{array}$$

Answer: $a_n=7\mathrm{due\; north}-22$

Try this! Find an equation for the general term of the given arithmetics sequence and apply it to calculate its 100^th term: $\frac{\mathrm{three}}{2},2,\frac{\mathrm{v}}{2},3,\frac{7}{2},\dots$

Reply: $a_{\mathrm{due\; north}}=\frac{\mathrm{ane}}{2}n+\mathrm{i}$ ; $a_{100}=51$

Arithmetic Serial

An arithmetic seriesThe sum of the terms of an arithmetic sequence. is the sum of the terms of an arithmetic sequence. For example, the sum of the commencement 5 terms of the sequence defined past $a_n=\mathrm{ii}n-1$ follows:

$$\begin{array}{ccc}\hfill {\mathrm{Due\; south}}_{\mathrm{v}}& =& {\displaystyle \underset{\mathrm{due\; north}=1}{\overset{5}{\Sigma }}\left(\mathrm{ii}\mathrm{northward}-1\right)}\hfill \\ & =& \left[\mathrm{ii}\left(\mathrm{i}\right)-1\right]+\left[2\left(2\right)-1\right]+\left[2\left(3\right)-1\right]+\left[2\left(4\right)-1\right]+\left[2\left(5\right)-1\right]\hfill \\ & =& 1+\mathrm{iii}+5+\mathrm{seven}+\mathrm{ix}\hfill \\ & =& 25\hfill \end{array}$$

Adding 5 positive odd integers, as we have washed above, is managable. However, consider calculation the first 100 positive odd integers. This would be very tedious. Therefore, nosotros adjacent develop a formula that can be used to summate the sum of the first n terms, denoted ${\mathrm{South}}_n$ , of whatever arithmetic sequence. In general,

$$S_{\mathrm{northward}}=a_1+\left(a_1+d\right)+\left(a_{\mathrm{ane}}+\mathrm{two}d\right)+\dots +a_n$$

Writing this series in reverse we accept,

$$S_{\mathrm{due\; north}}=a_n+\left(a_{\mathrm{due\; north}}-d\right)+\left(a_{\mathrm{due\; north}}-2d\right)+\dots +a_1$$

And adding these two equations together, the terms involving d add to nil and we obtain n factors of $a_1+a_{\mathrm{north}}$ :

$$\begin{array}{ccc}\hfill \mathrm{two}{S}_n& =& \left(a_1+a_n\right)+\left(a_{\mathrm{one}}+a_{\mathrm{north}}\right)+\dots +\left(a_n+a_1\right)\hfill \\ \hfill 2S_{\mathrm{north}}& =& n\left(a_1+a_n\right)\hfill \end{array}$$

Dividing both sides by ii leads us the formula for the nth partial sum of an arithmetics sequenceThe sum of the kickoff northward terms of an arithmetics sequence given by the formula: $S_{\mathrm{north}}=\frac{n\left(a_1+a_n\right)}{2}.$ :

$$S_n=\frac{n\left(a_{\mathrm{one}}+a_n\right)}{\mathrm{two}}$$

Utilise this formula to calculate the sum of the outset 100 terms of the sequence defined by $a_{\mathrm{due\; north}}=2n-1.$ Hither $a_1=1$ and $a_{100}=199.$

$$\begin{array}{ccc}\hfill {\mathrm{Due\; south}}_{100}& =& \frac{100\left(a_1+a_{100}\right)}{2}\hfill \\ & =& \frac{100\left(\mathrm{one}+199\right)}{2}\hfill \\ & =& \mathrm{10,000}\hfill \end{array}$$

Example 5

Find the sum of the first fifty terms of the given sequence: iv, 9, fourteen, xix, 24, …

Solution:

Determine whether or not there is a mutual difference between the given terms.

$$d=\mathrm{nine}-\mathrm{four}=5$$

Notation that the difference betwixt any two successive terms is 5. The sequence is indeed an arithmetics progression and we tin can write

$$\begin{array}{ccc}\hfill a_{\mathrm{due\; north}}& =& a_1+\left(n-1\right)d\hfill \\ & =& 4+\left(n-\mathrm{one}\right)\cdot 5\hfill \\ & =& 4+5n-\mathrm{five}\hfill \\ & =& 5\mathrm{north}-1\hfill \end{array}$$

Therefore, the general term is $a_n=\mathrm{v}n-\mathrm{ane}.$ To calculate the 50^thursday partial sum of this sequence nosotros need the 1^st and the 50^th terms:

$$\begin{array}{ccc}\hfill a_1& =& 4\hfill \\ \hfill a_{50}& =& \mathrm{five}\left(50\right)-1=249\hfill \end{array}$$

Next use the formula to determine the 50^thursday fractional sum of the given arithmetic sequence.

$$\begin{array}{ccc}\hfill {\mathrm{South}}_n& =& \frac{n(a_{\mathrm{one}}+a_{\mathrm{northward}})}{\mathrm{two}}\hfill \\ \hfill {\mathrm{Due\; south}}_{50}& =& \frac{50.(a_1+a_{50})}{\mathrm{ii}}\hfill \\ & =& \frac{50(\mathrm{four}+249)}{2}\hfill \\ & =& 25(253)\hfill \\ & =& \mathrm{half\; dozen,325}\hfill \end{array}$$

Answer: $S_{\mathrm{fifty}}=\mathrm{6,325}$

Example half dozen

Evaluate: ${\displaystyle \underset{n=1}{\overset{35}{\Sigma }}\left(10-4n\right)}$ .

Solution:

In this case, nosotros are asked to find the sum of the start 35 terms of an arithmetic sequence with general term $a_n=\mathrm{x}-4n.$ Use this to determine the ane^st and the 35^th term.

$$\begin{array}{ccc}\hfill a_1& =& 10-\mathrm{four}\left(1\right)=\mathrm{half-dozen}\hfill \\ \hfill a_{35}& =& 10-\mathrm{iv}\left(35\right)=-130\hfill \end{array}$$

Next use the formula to determine the 35^th partial sum.

$$\begin{array}{ccc}\hfill S_{\mathrm{due\; north}}& =& \frac{\mathrm{due\; north}\left(a_{\mathrm{ane}}+a_n\right)}{2}\hfill \\ \hfill S_{35}& =& \frac{35\cdot \left(a_{\mathrm{i}}+a_{35}\right)}{\mathrm{ii}}\hfill \\ & =& \frac{35\left[6+\left(-130\right)\right]}{2}\hfill \\ & =& \frac{35\left(-124\right)}{\mathrm{two}}\hfill \\ & =& -\mathrm{2,170}\hfill \end{array}$$

Answer: −2,170

Example 7

The first row of seating in an outdoor amphitheater contains 26 seats, the 2d row contains 28 seats, the third row contains 30 seats, and and so on. If in that location are eighteen rows, what is the full seating capacity of the theater?

Figure 9.ii

Roman Theater (Wikipedia)

Solution:

Begin by finding a formula that gives the number of seats in any row. Here the number of seats in each row forms a sequence:

$$26,28,30,\dots$$

Notation that the divergence between whatever ii successive terms is two. The sequence is an arithmetics progression where $a_{\mathrm{ane}}=26$ and $d=2.$

$$\begin{array}{ccc}\hfill a_n& =& a_1+\left(n-1\right)d\hfill \\ & =& 26+\left(\mathrm{northward}-1\right)\cdot \mathrm{ii}\hfill \\ & =& 26+2n-2\hfill \\ & =& 2n+24\hfill \end{array}$$

Therefore, the number of seats in each row is given by $a_{\mathrm{northward}}=\mathrm{two}n+24.$ To calculate the total seating capacity of the 18 rows we need to summate the 18^th fractional sum. To practise this we demand the 1^st and the 18^th terms:

$$\begin{array}{ccc}\hfill a_{\mathrm{i}}& =& 26\hfill \\ \hfill a_{\mathrm{eighteen}}& =& 2\left(18\right)+24=\mathrm{lx}\hfill \end{array}$$

Use this to calculate the 18^th fractional sum every bit follows:

$$\begin{array}{ccc}{S}_n\hfill & =\hfill & \frac{\mathrm{northward}\left(a_{\mathrm{i}}+a_n\right)}{2}\hfill \\ \hfill {\mathrm{Southward}}_{18}& =& \frac{18\cdot \left(a_1+a_{18}\right)}{2}\hfill \\ & =& \frac{\mathrm{xviii}\left(26+60\right)}{2}\hfill \\ & =& \mathrm{nine}\left(86\right)\hfill \\ & =& 774\hfill \end{array}$$

Respond: At that place are 774 seats full.

Endeavor this! Find the sum of the showtime sixty terms of the given sequence: v, 0, −five, −x, −15, …

Answer: ${\mathrm{South}}_{\mathrm{threescore}}=-\mathrm{eight,550}$

Key Takeaways

• An arithmetic sequence is a sequence where the difference d between successive terms is abiding.
• The full general term of an arithmetic sequence can be written in terms of its first term $a_{\mathrm{ane}}$ , common deviation d, and index northward as follows: $a_n=a_{\mathrm{i}}+\left(n-1\right)d.$
• An arithmetic serial is the sum of the terms of an arithmetic sequence.
• The nth partial sum of an arithmetic sequence can be calculated using the first and last terms as follows: $S_{\mathrm{northward}}=\frac{n\left(a_1+a_n\right)}{\mathrm{ii}}.$

Topic Exercises

Part A: Arithmetic Sequences

Write the first 5 terms of the arithmetics sequence given its first term and common difference. Discover a formula for its general term.

1. $a_{\mathrm{one}}=5$ ; $d=3$

2. $a_{\mathrm{one}}=12$ ; $d=2$

3. $a_1=15$ ; $d=-\mathrm{v}$

4. $a_1=\mathrm{seven}$ ; $d=-4$

5. $a_1=\frac{1}{2}$ ; $d=1$

6. $a_1=\frac{2}{3}$ ; $d=\frac{1}{3}$

7. $a_1=1$ ; $d=-\frac{1}{\mathrm{two}}$

8. $a_1=-\frac{5}{4}$ ; $d=\frac{1}{\mathrm{iv}}$

9. $a_{\mathrm{ane}}=\mathrm{ane.8}$ ; $d=\mathrm{0.six}$

10. $a_{\mathrm{one}}=-4.3$ ; $d=\mathrm{2.i}$

Given the arithmetic sequence, detect a formula for the general term and use it to decide the 100^th term.

11. iii, 9, 15, 21, 27,…

12. 3, 8, 13, eighteen, 23,…

13. −3, −7, −11, −15, −xix,…

14. −6, −xiv, −22, −thirty, −38,…

15. −5, −10, −fifteen, −twenty, −25,…

16. 2, 4, 6, 8, 10,…

17. $\frac{1}{\mathrm{ii}}$ , $\frac{\mathrm{five}}{2}$ , $\frac{9}{\mathrm{ii}}$ , $\frac{13}{\mathrm{ii}}$ , $\frac{17}{2}$ ,…

18. $-\frac{1}{\mathrm{iii}}$ , $\frac{2}{3}$ , $\frac{5}{3}$ , $\frac{\mathrm{viii}}{3}$ , $\frac{11}{3}$ ,…

19. $\frac{\mathrm{i}}{3}$ , 0, $-\frac{1}{3}$ , $-\frac{2}{3}$ , −1,…

20. $\frac{\mathrm{i}}{\mathrm{iv}}$ , $-\frac{\mathrm{one}}{2}$ , $-\frac{5}{4}$ , −ii, $-\frac{11}{4}$ ,…

21. 0.eight, two, 3.ii, 4.iv, five.6,…

22. 4.four, 7.5, 10.six, 13.7, xvi.8,…

23. Observe the l^th positive odd integer.

24. Find the l^th positive even integer.

25. Discover the twoscore^th term in the sequence that consists of every other positive odd integer: 1, 5, 9, 13,…

26. Observe the 40^th term in the sequence that consists of every other positive even integer: 2, 6, 10, 14,…

27. What number is the term 355 in the arithmetics sequence −15, −5, 5, 15, 25,…?

28. What number is the term −172 in the arithmetics sequence four, −4, −12, −20, −28,…?

29. Given the arithmetic sequence divers by the recurrence relation $a_{\mathrm{north}}=a_{n-1}+5$ where $a_1=2$ and $n>1$ , notice an equation that gives the general term in terms of $a_1$ and the common difference d.

30. Given the arithmetics sequence defined by the recurrence relation $a_{\mathrm{due\; north}}=a_{n-1}-9$ where $a_{\mathrm{ane}}=4$ and $n>1$ , discover an equation that gives the full general term in terms of $a_{\mathrm{one}}$ and the mutual difference d.

Given the terms of an arithmetics sequence, find a formula for the general term.

31. $a_{\mathrm{i}}=\mathrm{vi}$ and $a_{\mathrm{vii}}=42$

32. $a_1=-\frac{\mathrm{i}}{2}$ and $a_{12}=-6$

33. $a_1=-\mathrm{nineteen}$ and $a_{26}=56$

34. $a_1=-\mathrm{nine}$ and $a_{31}=141$

35. $a_{\mathrm{ane}}=\frac{1}{\mathrm{half\; dozen}}$ and $a_{10}=\frac{37}{\mathrm{half-dozen}}$

36. $a_{\mathrm{one}}=\frac{\mathrm{five}}{4}$ and $a_{11}=\frac{65}{4}$

37. $a_3=6$ and $a_{26}=-\mathrm{forty}$

38. $a_3=16$ and $a_{15}=76$

39. $a_4=-8$ and $a_{23}=30$

40. $a_5=-7$ and $a_{37}=-135$

41. $a_{\mathrm{four}}=-\frac{23}{\mathrm{ten}}$ and $a_{21}=-\frac{25}{2}$

42. $a_3=\frac{1}{\mathrm{eight}}$ and $a_{12}=-\frac{11}{\mathrm{two}}$

43. $a_5=\mathrm{thirteen.2}$ and $a_{26}=61.5$

44. $a_4=-1.2$ and $a_{13}=\mathrm{12.iii}$

Discover all arithmetic ways between the given terms.

45. $a_1=-\mathrm{three}$ and $a_6=17$

46. $a_{\mathrm{ane}}=5$ and $a_{\mathrm{five}}=-7$

47. $a_{\mathrm{ii}}=4$ and $a_8=\mathrm{seven}$

48. $a_5=\frac{\mathrm{i}}{2}$ and $a_{\mathrm{nine}}=-\frac{7}{2}$

49. $a_{\mathrm{five}}=\mathrm{fifteen}$ and $a_7=21$

50. $a_{\mathrm{six}}=4$ and $a_{11}=-\mathrm{ane}$

Part B: Arithmetic Series

Summate the indicated sum given the formula for the full general term.

51. $a_{\mathrm{northward}}=3\mathrm{north}+\mathrm{v}$ ; ${\mathrm{South}}_{100}$

52. $a_{\mathrm{northward}}=\mathrm{v}\mathrm{north}-\mathrm{xi}$ ; $S_{100}$

53. $a_n=\frac{1}{\mathrm{ii}}-n$ ; $S_{70}$

54. $a_n=1-\frac{\mathrm{iii}}{2}n$ ; $S_{120}$

55. $a_n=\frac{\mathrm{i}}{2}\mathrm{northward}-\frac{3}{4}$ ; $S_{20}$

56. $a_n=n-\frac{3}{\mathrm{five}}$ ; $S_{150}$

57. $a_{\mathrm{northward}}=45-5n$ ; ${\mathrm{Due\; south}}_{65}$

58. $a_{\mathrm{due\; north}}=2n-48$ ; $S_{95}$

59. $a_n=4.4-1.6n$ ; $S_{75}$

60. $a_n=6.5n-3.3$ ; $S_{67}$

Evaluate.

61. $${\displaystyle \sum _{\mathrm{northward}=1}^{160}\left(3n\right)}$$

62. $${\displaystyle \sum _{\mathrm{northward}=\mathrm{one}}^{121}\left(-\mathrm{two}n\right)}$$

63. $${\displaystyle \sum _{n=1}^{250}\left(4\mathrm{due\; north}-3\right)}$$

64. $${\displaystyle \sum _{\mathrm{due\; north}=\mathrm{one}}^{120}\left(\mathrm{ii}n+12\right)}$$

65. $${\displaystyle \sum _{n=\mathrm{ane}}^{70}\left(19-8\mathrm{due\; north}\right)}$$

66. $${\displaystyle \sum _{n=1}^{220}\left(5-n\right)}$$

67. $${\displaystyle \sum _{n=1}^{60}\left(\frac{5}{\mathrm{ii}}-\frac{\mathrm{one}}{\mathrm{two}}n\right)}$$

68. $${\displaystyle \sum _{n=1}^{51}\left(\frac{3}{8}\mathrm{north}+\frac{\mathrm{ane}}{4}\right)}$$

69. $${\displaystyle \sum _{\mathrm{north}=1}^{120}\left(1.5n-2.6\right)}$$

70. $${\displaystyle \sum _{n=1}^{175}\left(-0.2n-1.6\right)}$$

71. Find the sum of the first 200 positive integers.

72. Notice the sum of the first 400 positive integers.

The general term for the sequence of positive odd integers is given past $a_{\mathrm{due\; north}}=2n-\mathrm{ane}$ and the general term for the sequence of positive even integers is given by $a_n=2\mathrm{due\; north}.$ Detect the post-obit.

73. The sum of the offset 50 positive odd integers.

74. The sum of the kickoff 200 positive odd integers.

75. The sum of the first 50 positive even integers.

76. The sum of the first 200 positive even integers.

77. The sum of the outset k positive odd integers.

78. The sum of the get-go k positive even integers.

79. The kickoff row of seating in a small theater consists of eight seats. Each row thereafter consists of 3 more seats than the previous row. If there are 12 rows, how many total seats are in the theater?

80. The first row of seating in an outdoor amphitheater contains 42 seats, the 2d row contains 44 seats, the third row contains 46 seats, and so on. If there are 22 rows, what is the total seating capacity of the theater?

81. If a triangular stack of bricks has 37 bricks on the bottom row, 34 bricks on the second row so on with i brick on top. How many bricks are in the stack?

82. Each successive row of a triangular stack of bricks has one less brick until there is but i brick on superlative. How many rows does the stack have if in that location are 210 total bricks?

83. A 10-year salary contract offers $65,000 for the first year with a $3,200 increase each additional year. Make up one's mind the total salary obligation over the 10 year catamenia.

84. A clock belfry strikes its bell the number of times indicated by the hr. At i o'clock it strikes once, at two o'clock it strikes twice then on. How many times does the clock tower strike its bell in a twenty-four hours?

Part C: Discussion Board

85. Is the Fibonacci sequence an arithmetics sequence? Explain.

86. Employ the formula for the nth fractional sum of an arithmetics sequence ${\mathrm{Due\; south}}_n=\frac{n\left(a_{\mathrm{one}}+a_n\right)}{\mathrm{ii}}$ and the formula for the general term $a_n=a_1+\left(n-1\right)d$ to derive a new formula for the due northth partial sum $S_n=\frac{n}{\mathrm{two}}\left[\mathrm{ii}{a}_{\mathrm{i}}+\left(n-\mathrm{ane}\right)d\right].$ Under what circumstances would this formula be useful? Explicate using an example of your own making.

87. Discuss methods for calculating sums where the index does not start at 1. For case, ${\displaystyle \underset{n=15}{\overset{35}{\Sigma }}\left(\mathrm{three}n+4\right)}=\mathrm{i,659}.$

88. A famous story involves Carl Friedrich Gauss misbehaving at schoolhouse. As punishment, his teacher assigned him the task of adding the outset 100 integers. The legend is that immature Gauss answered correctly within seconds. What is the respond and how practice you think he was able to find the sum so rapidly?

Answers

1. 5, 8, 11, 14, 17; $a_n=3n+2$

3. 15, x, five, 0, −5; $a_n=20-5n$

5. $\frac{\mathrm{ane}}{2}$ , $\frac{3}{\mathrm{ii}}$ , $\frac{5}{\mathrm{two}}$ , $\frac{\mathrm{seven}}{\mathrm{two}}$ , $\frac{9}{2}$ ; $a_{\mathrm{north}}=\mathrm{northward}-\frac{1}{\mathrm{ii}}$

7. 1, $\frac{\mathrm{ane}}{\mathrm{ii}}$ , 0, $-\frac{1}{2}$ , −i; $a_n=\frac{\mathrm{iii}}{2}-\frac{\mathrm{one}}{2}\mathrm{north}$

9. 1.8, 2.4, three, iii.six, four.2; $a_n=0.6n+\mathrm{one.2}$

11. $a_n=\mathrm{vi}n-\mathrm{iii}$ ; $a_{100}=597$

13. $a_n=\mathrm{ane}-4n$ ; $a_{100}=-399$

15. $a_{\mathrm{northward}}=-\mathrm{five}n$ ; $a_{100}=-500$

17. $a_{\mathrm{northward}}=2n-\frac{3}{\mathrm{two}}$ ; $a_{100}=\frac{397}{\mathrm{ii}}$

19. $a_n=\frac{2}{3}-\frac{1}{3}n$ ; $a_{100}=-\frac{98}{3}$

21. $a_{\mathrm{due\; north}}=1.2n-0.4$ ; $a_{100}=119.6$

23. 99

25. 157

27. 38

29. $a_{\mathrm{due\; north}}=5n-3$

31. $a_n=6n$

33. $a_n=3n-22$

35. $a_n=\frac{2}{3}n-\frac{1}{2}$

37. $a_n=12-2\mathrm{northward}$

39. $a_{\mathrm{north}}=2n-\mathrm{xvi}$

41. $a_n=\frac{1}{10}-\frac{3}{5}n$

43. $a_n=2.3n+\mathrm{1.vii}$

45. 1, v, 9, 13

47. $\frac{9}{\mathrm{two}}$ , five, $\frac{11}{\mathrm{two}}$ , 6, $\frac{\mathrm{thirteen}}{2}$

49. 18

51. 15,650

53. −two,450

55. 90

57. −7,800

59. −4,230

61. 38,640

63. 124,750

65. −xviii,550

67. −765

69. 10,578

71. xx,100

73. 2,500

75. 2,550

77. ${\mathrm{grand}}^{\mathrm{two}}$

79. 294 seats

81. 247 bricks

83. $794,000

85. Answer may vary

87. Answer may vary

What Is The Sum Of The Terms Of The Series (Use The Formula): 7 11 15 . . . 35,

Source: https://saylordotorg.github.io/text_intermediate-algebra/s12-02-arithmetic-sequences-and-serie.html

Posted by: freyfraidgetefe.blogspot.com

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