The sequence 5, 17, 65 has as its nth term \({4^n} + 1\). Find

(i)  the fourth term.

(ii)  the sum of the first \(n\) terms.

For the geometric progression \(2,\,\,\dfrac{1}{2},\,\,\dfrac{1}{8},\,\,...\)

(i)  Find the sum of the first \(n\) terms

(ii)  Find the least value of \(n\) for which the sum exceeds 2.66

The 6th term of a geometric progression is 16 and the 3rd term is 2.

(i)  Find the common ratio.

(ii)  Find the first term.

A geometric series has a second term 4.5 and the ratio of the fourth term to the third term is 1.5.

i) Find the common ratio \(r\).

ii) What is the first term \(a\)?

iii) Calculate the sum of the first 9 terms.

The sum of the first three terms of a geometric progression is 152 and the sum of the next three terms 513. Find

i) the common ratio.

ii) the first term.

The sum of the first eight terms or the G.P. \(5+15+\) ... is?

The sum of the first 10 terms of the geometric series \(3 + 6 + 12 + ...\) is?

Find the number of terms in the series 3 + 6 + 12 + ... whose sum is 381

How many terms of the G.P. \(2-6+18-54\) ... are needed to make a total of \(\dfrac{1}{2}(1 - {3^8})\)

The 4th term of a G.P.  -6 and the 7th term is 48. The sum of the first seven terms is?

The sum of the first two terms of a G.P. is 3, and the sum of the 2nd and 3rd terms is -6. Find the common ratio.

How many terms of the G.P. \(1+3+9+\)... are required to make a total of more than a million?

Which of the following is the correct value \(\mathrm{S}_n\) for the series given below?

\[\mathrm{S}_n=5+15+45+\ldots+98415\]