Match List-I with List-II (Symbols have their usual meaning)
| List-I | List-II | ||
|---|---|---|---|
| Expression | Definitions | ||
| (A) | \(a_{n}=a+[n-1]d\) | (I) | nth term of Geometric Progression |
| (B) | \(s_{n}=\displaystyle\frac{n}{2}[2a+(n-1)d]\) | (II) | Sum of n terms of Geometric Progression |
| (C) | \(a_{n}=ar^{n-1}\) | (III) | nth term of Arithmetic Progression |
| (D) | \(s_{n}=a\left(\displaystyle\frac{r^{n}-1}{r-1}\right)\) | (IV) | Sum of n terms of Arithmetic Progression |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) - (III), (B) - (IV), (C) - (I), (D) - (II)
In an arithmetic progression (AP) each term is the one before it plus a fixed difference \(d\); in a geometric progression (GP) each term is the one before it times a fixed ratio \(r\). So the formulas with \(d\) belong to the AP and those with \(r\) to the GP.
(A) \(a_{n}=a+[n-1]d\) — this matches (III) nth term of Arithmetic Progression. Starting from \(a\), the difference \(d\) is added \(n-1\) times to reach the \(n\)th term.
(B) \(s_{n}=\frac{n}{2}[2a+(n-1)d]\) — this matches (IV) Sum of n terms of Arithmetic Progression. It is \(n\) times the average of the first and the last term, \(a\) and \(a+(n-1)d\).
(C) \(a_{n}=ar^{n-1}\) — this matches (I) nth term of Geometric Progression. Starting from \(a\), the term is multiplied by \(r\) another \(n-1\) times.
(D) \(s_{n}=a\left(\frac{r^{n}-1}{r-1}\right)\) — this matches (II) Sum of n terms of Geometric Progression, the standard sum of a GP when \(r\ne1\).