Q222 / 270 · NDA 2023 (Phase 2) · Mathematics

Let $a_1, a_2, a_3 \ldots$ be in AP such that $a_1+a_5+a_{10}+a_{15}+a_{20}+a_{25}+a_{30}+a_{34}=300$ What is $\sum_{n=1}^{34} a_n$ equal to?

  • A
    900
  • B
    1025
  • C
    1200
  • Correct answer
    D
    1275

Solution

Given, $\begin{aligned} & a_1+a_5+a_{10}+a_{15}+a_{20}+a_{25}+a_{30}+a_{34}=300 \\ & =a_1+a_1+4 d+a_1+9 d+a_1+14 d+a_1+19 d \\ & +a_1+24 d+a_1+29 d+a_1+33 d=300 \\ & =8 a_1+132 d=300 \\ & =\left(2 a_1+33 d\right)=\frac{300}{4}...(i) \end{aligned}$ $\begin{aligned} & \text { Since, } \sum_{n=1}^{34} a_n=a_1+a_2+\ldots \ldots \ldots+a_{34} \\ & =\frac{34}{2}\left(2 a_1+33 d\right) \\ & =17 \times \frac{300}{}=1275 \end{aligned}$

Hence, the correct answer is option: 'D'.

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