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<sup>th</sup> term is 2n. The common difference d is:
According to the question, \( \mathrm{T}_{n} = 2n, \, a = n \) \[ 2n = n + (n - 1)d \] \[ \Rightarrow n = (n - 1)d \Rightarrow d = \frac{n}{n - 1} \] We have \[ \frac{n}{2}[2n + (n - 1)d] = 216 \] \[ \Rightarrow \frac{n}{2}[3n] = 216 \Rightarrow \frac{3}{2}n^2 = 216 \] \[ \Rightarrow n^2 = \frac{2 \times 216}{3} = 2 \times 72 = 144 \] \[ \Rightarrow n = 12 \Rightarrow d = \frac{12}{11} \]
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