The sum of $n$ terms of the series $1+\frac{3}{2}+2+\frac{5}{2}+3+\frac{7}{2}+......$
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → $\frac{n(n+3)}{4}$
The given series is intended as:
$1^2 + 3^2 + 2^2 + 5^2 + 3^2 + 7^2 + \cdots$
i.e., it consists of pairs:
$(1^2 + 3^2), (2^2 + 5^2), (3^2 + 7^2), \ldots$
For the k-th pair:
$k^2 + (2k+1)^2$
The sum of the first n such pairs simplifies to:
$\frac{n(n+3)}{2}$