Which of the following statement(s) is/are correct? Here $$a$$ and $$r$$ have their usual meaning:

(A) $$a, ar, ar^2, \ldots$$ is a G.P.
(B) $$n^{th}$$ term of a G.P. is $$a_n = ar^{n-t}$$.
(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 $$S_n = a \left(\frac{r^n - 1}{r - 1} ight), r
eq 1$$.
(E) $$4, 16, 256, \ldots$$ is a G.P.

Choose the correct answer from the options given below:
(a) (A) and (B) only
(b) (C) and (D) only
(c) (A), (B), (D), and (E) only
(d) (A), (B), and (D) only

Question

Which of the following statement(s) is/are correct? Here $$a$$ and $$r$$ have their usual meaning:

(A) $$a, ar, ar^2, \ldots$$ is a G.P.  
(B) $$n^{th}$$ term of a G.P. is $$a_n = ar^{n-t}$$.  
(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 $$S_n = a \left(\frac{r^n - 1}{r - 1}ight), r 
eq 1$$.  
(E) $$4, 16, 256, \ldots$$ is a G.P.

Choose the correct answer from the options given below:  
(a) (A) and (B) only  
(b) (C) and (D) only  
(c) (A), (B), (D), and (E) only  
(d) (A), (B), and (D) only

Accepted Answer

1. **Analyze each statement:**  
   - **(A):** $$a, ar, ar^2, \ldots$$ forms a geometric progression because each term is multiplied by the common ratio $$r$$. This is correct.  
   - **(B):** The formula for the $$n^{th}$$-term of a G.P. is $$a_n = ar^{n-t}$$. This matches the standard formula for terms in G.P., so it is correct.  
   - **(C):** The formula for the $$n^{th}$$-term given as $$a_n = ar^{n+1}$$ is incorrect because it does not align with the standard formula for terms in G.P.  
   - **(D):** The sum of the first $$n$$-terms of a G.P. is correctly given as $$S_n = a \left(\frac{r^n - 1}{r - 1}ight)$$, where $$r 
eq 1$$. This is correct.  
   - **(E):** The sequence $$4, 16, 256, \ldots$$ does not form a G.P., as the common ratio between consecutive terms varies. This is incorrect.

2. **Conclusion:**  
   - Statements **(A)**, **(B)**, and **(D)** are correct.

Thus, the correct answer is **(d): (A), (B), and (D)**.

Answer

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