NIMCET 2013 — Mathematics PYQ

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Question

NIMCET 2013 — Mathematics PYQ

NIMCET | Mathematics | 2013

The sum of first $n$ terms of an arithmetic series is $216$ . The value of the first term is $n$ and the value of the $n^{\text{th}}$ term is $2n$ . The common difference $d$ is:

Choose the correct answer:

Explanation

Solution

Given:

Sum of $n$ terms $(S_n) = 216$
First term $(a) = n$
Last term ( $l$ or $a_n$ ) $= 2n$

1. Find the value of $n$ :

Using the formula $S_n = \frac{n}{2}(a + l)$ :

216 = \frac{n}{2}(n + 2n)

216 = \frac{n}{2}(3n)

216 = \frac{3n^2}{2}

432 = 3n^2

n^2 = \frac{432}{3}

n^2 = 144

n = 12

2. Find the common difference $d$ :

Now, $a = n = 12$ and $a_n = 2n = 24$ .

Using the formula $a_n = a + (n - 1)d$ :

24 = 12 + (12 - 1)d

24 - 12 = 11d

12 = 11d

d = \frac{12}{11}