Multiple choice

If the $n^{th}$ term of the series is $T_n = 2n -1$, then sum of $n$ terms $S_n$ will be

  1. $n^2$
  2. $\dfrac {n(n+1)}{2}$
  3. $\dfrac {n(n+1)(n+2)}{6}$
  4. $n+2$
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A Correct answer
Explanation

The series is 1, 3, 5, ..., (2n-1). This is an arithmetic progression with first term a=1 and last term l=2n-1. The sum is (n/2)(a + l) = (n/2)(1 + 2n - 1) = (n/2)(2n) = n^2.

AI explanation

Using the formula for the sum of the first n terms of an arithmetic progression, Sn = n/2 * (a + l), where the first term a is 1 and the last term is 2n - 1. Substituting these values gives Sn = n/2 * (1 + 2n - 1), which simplifies to n/2 * 2n. This further reduces to n * n, resulting in n^2.

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