Sequence & Series Solutions

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

1. The next two terms are 25, 30 and.
2. The next two terms are 1/5, 1/6 and.
3. The next two terms are 35, 48 and.
4. The next two terms are 1/30, 1/42 and.

2. Write the following series in sigma notation

3. Write down the first three terms, and where there is one, the last term of each of the following series

1. 
2. 0+2+6+…+30
3. 
4. 
5. 

4. Write down the nth term and the stated term of the following A.P's.

The nth term of an A.P. is

5. Find the sums of the following series

The sum of an A.P. with n terms, where  (this is a rearrangement of the formula for the last term)

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

 and so.

2. Here a=-17, l=33 and d=5. So  and .
3. a=1, l=9 ¼ and d= ¼ and so .

8. 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, therefore.

Rearranging the formula  to make d the subject, we obtain.

9. 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 that, where  and. Therefore,.

10. 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,.

Also

The numbers are 8, 11 and 14.

11. 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 that, a=12 and d=3.

Hence, using

and

Equating the sums

The non-zero value of n is n=7.

12. Write down the

20. 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

r=0.8 and S=250, thus

The fourth term is given by.

21. The sum of the first n terms of a geometric series is. 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

The first term is.

The second term is.

Thus,.

The sum to infinity is.

22. 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 and.

The sum of n terms is.

The value of k is the smallest number bigger than for all values of n. Thus, .

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

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

23. Show that