UPSC NDA Math Model Paper 10

Let the nth term and sum of the series upto n terms of the series be $T_n$ and $S_n$ , respectively. Then, ⇒$S_n =\frac{1}{5}+\frac{3}{5}+1+\frac{7}{5}+\frac{9}{5}+\dots$ Multiplying the above equation by 5, we get $\Rightarrow 5S_n = 1 + 3 + 5 + 7 + 9 + \dots$ up to n terms This is the sum of first odd natural numbers. As we know that, sum of the first n odd natural numbers i.e 1 + 3 + 5 +7 +9 + ...up to n terms = n ⇒ $\Rightarrow 5 S_n = n^{2} \Rightarrow S_n = \frac{n^{2}}{5}$ For $n=50, S_{50}=\frac{50^{2}}{5}=500$. Hence, option 2 is the correct answer.