Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.
Step-by-Step Solution
Key Concept: Use the nth term formula of an AP, $a_n = a + (n-1)d$, to find the first term $a$. Then apply the sum formula $S_n = \frac{n}{2}[2a + (n-1)d]$ (or $S_n = \frac{n}{2}(a + l)$) to obtain the required sum.
1. Given data: common difference $d = 7$, number of terms $n = 22$, 22nd term $a_{22} = 149$.
2. Find the first term $a$ using the nth‑term formula:
$$a_{22} = a + (22-1)d \Rightarrow 149 = a + 21\times7$$
$$a = 149 - 147 = 2.$$
3. Find the sum of the first 22 terms. Two equivalent formulas can be used:
- Using $a$ and $d$:
$$S_{22} = \frac{22}{2}\big[2a + (22-1)d\big]$$
$$= 11\big[2\times2 + 21\times7\big]$$
$$= 11\big[4 + 147\big] = 11\times151 = 1661.$$
- Using first and last terms ($a$ and $l = a_{22}$):
$$S_{22} = \frac{22}{2}(a + l) = 11(2 + 149) = 11\times151 = 1661.$$
4. Result: The sum of the first 22 terms of the AP is $\boxed{1661}$.
Correct Answer: 1661