Which of the following statement(s) is/are correct? Here \(a\) and \(r\) have their usual meanings:
(A) \(a, ar, ar^2, ....\) is a G.P.
(B) \(n^{th}\) term of a G.P. is \(a_{n}=ar^{n-1}\)
(C) \(n^{th}\) term of a G.P. is \(a_{n}=ar^{n+1}\)
(D) Sum of first \(n\) terms of a G.P. is \(S_n=a\left(\frac{r^n-1}{r-1}\right), r \neq 1\)
(E) \(1,4,16,256, ....\) is a G.P.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (D) only
A geometric progression (G.P.) is a sequence in which each term is the previous term multiplied by the same number, the common ratio \(r\); \(a\) is the first term.
(A) \(a, ar, ar^2, \ldots\) is a G.P. Correct. Each term is the one before it multiplied by \(r\), which is the definition of a G.P.
(B) \(n^{th}\) term of a G.P. is \(a_n=ar^{n-1}\) Correct. The first term is \(ar^0\), the second \(ar^1\), so the \(n\)th term carries the power \(n-1\).
(C) \(n^{th}\) term of a G.P. is \(a_n=ar^{n+1}\) Incorrect. For the first term this would give \(ar^2\) instead of \(a\); the power must be \(n-1\), not \(n+1\).
(D) Sum of first \(n\) terms of a G.P. is \(S_n=a\left(\frac{r^n-1}{r-1}\right), r\ne1\) Correct. This is the standard sum formula; \(r\ne1\) is needed because for \(r=1\) the denominator would be zero (the sum is then \(na\)).
(E) \(1, 4, 16, 256, \ldots\) is a G.P. Incorrect. The ratios of each term to the one before are 4, 4 and then 16; the ratio is not constant (a G.P. would have 64 after 16).