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Sequence & Series

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Introduction

Sequence & Series Solutions

  1. For the following sequences write down the next two terms of the sequence and an expression for the nth term.

a. 5, 10, 15, 20, …

b. 1, 1/2, 1/3, 1/4, …

c. 0, 3, 8, 15, 24, …

d. 1/2, 1/6, 1/12, 1/20, …

  1. The next two terms are 25, 30 andimage00.png.
  2. The next two terms are 1/5, 1/6 andimage01.png.
  3. The next two terms are 35, 48 andimage48.png.
  4. The next two terms are 1/30, 1/42 andimage59.png.
  1. Write the following series in sigma notation

a. 1+8+27+64+125

b. 2+4+6+8+…+20

c. image70.png

d. image80.png

e. –4-1+2+5+…+17

f. image91.png

  1. image101.png
  2. image112.png
  3. image119.png
  4. image02.png
  5. image13.png
  6. image24.png
  1. Write down the first three terms, and where there is one, the last term of each of the following series

a. image35.png

b. image42.png

c. image43.png

d. image44.png

e. image45.png

  1. image46.png
  2. 0+2+6+…+30
  3. image47.png
  4. image49.png
  5. image50.png
  1. Write down the nth term and the stated term of the following A.P's.

a. 7+11+15+… (7th)

b. –7-5-3-… (23rd)

The nth term of an A.P. is image51.png

  1. image52.png

image53.png

  1. image54.png

image55.png

  1. Find the sums of the following series

a. 5+9+13+…+101

b. –17-12-7-…+33

c. 1+ 1 ¼ + 1 ½ + … + 9 ¾

The sum of an A.P. with n termsimage56.png, where image57.png (this is a rearrangement of the formula for the last termimage58.png)

  1. It can be seen that a=5, l=101 and d=4, therefore

image60.png and soimage61.png.

  1. Here a=-17, l=33 and d=5. So image62.png and image63.png.
  2. a=1, l=9 ¼ and d= ¼ and so image64.png.
...read more.

Middle

, we obtainimage77.png.
  1. The first and last terms of an A.P. with 25 terms are 29 and 179. Find the sum of the series and the common difference.

a=29, l=179 and n=25, thereforeimage78.png.

Rearranging the formula image57.png to make d the subject, we obtainimage79.png.

  1. The sum of the first four terms of an A.P. is twice the 5th term. Show that the common difference is equal to the first term.

From the question we know thatimage81.png, where image82.png andimage83.png. Therefore,image84.png.

  1. Three numbers in an A.P have sum 33 and product 1232. Find the numbers.

Let the numbers be a, b and c.

From the definition of an A.P. we know that a=b-d and c=b+d.

Thus,image85.png.

Also

image86.png

image87.png

The numbers are 8, 11 and 14.

  1. In an A.P. the sum of the first 2n terms is equal to the sum of the next n terms. If the first term is 12 and the common difference is 3, find the non-zero value of n.

From the question we know thatimage88.png, a=12 and d=3.

Hence, using image89.png

image90.png

and

image92.png

Equating the sums

image93.png

The non-zero value of n is n=7.

  1. Write down the
...read more.

Conclusion

a=3 and image17.png. It is convergent.

image18.png

  1. A geometric series with common ratio 0.8 converges to the sum 250. Find the fourth term of the series.

The sum of a geometric series is image19.png

r=0.8 and S=250, thus

image20.png

The fourth term is given byimage21.png.

  1. The sum of the first n terms of a geometric series isimage22.png. Find the first term of the series, its common ratio and its sum to infinity.

If two adjacent terms of a G.P. are known then image23.png

The first term isimage25.png.

The second term isimage26.png.

Thus,image27.png.

The sum to infinity isimage28.png.

  1. Find the sum to n terms of the geometric series 108+60+33 1/3 +… If k is the least number which exceeds this sum for all values of n, find k. Find also the least value of n for which the sum exceeds 99% of k.

a=108 andimage29.png.

The sum of n terms isimage30.png.

The value of k is the smallest number bigger than image31.pngfor all values of n. Thus, image32.png.

The least value of n for which the sum exceeds 99% of k is given by the solution of

image33.png

image34.png

Taking logs and noticing that image36.png (this means that when we divide, the greater than sign is reversed)

image37.png

  1. Show that image38.png

image39.png

image40.png

image41.png

...read more.

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