Question

The sum of the first n terms of the AP whose first term is 1 and common difference is 2 is :

• A. 3n
• B. 2n - 1
• C. n^2
• D. n(n + 1)

Hint:

The AP $1, 3, 5, 7, \dots$ consists of odd numbers, and the sum of the first $n$ odd numbers is always $n^2$.

Answer 1

The Correct Option is C

Step 1: Identify the given AP.}
The arithmetic progression has first term $a = 1$ and common difference $d = 2$. So the AP is: \[ 1,\ 3,\ 5,\ 7,\ \dots \] This is the sequence of the first $n$ odd numbers.

Step 2: Use the formula for the sum of n terms of an AP.}
The formula for the sum of the first $n$ terms of an AP is: \[ S_n = \frac{n}{2}\left[2a + (n-1)d\right] \] Substituting $a = 1$ and $d = 2$, we get: \[ S_n = \frac{n}{2}\left[2(1) + (n-1)(2)\right] \] \[ S_n = \frac{n}{2}\left[2 + 2n - 2\right] \] \[ S_n = \frac{n}{2}(2n) \] \[ S_n = n^2 \]

Step 3: Compare with the given options.}

• (A) $3n$: Incorrect. This is not the sum obtained from the AP formula.
• (B) $2n - 1$: Incorrect. This gives the $n$th term of the AP, not the sum.
• (C) $n^2$: Correct. The sum of the first $n$ odd numbers is always $n^2$.
• (D) $n(n + 1)$: Incorrect. This is not the required sum for this AP.

Step 4: Conclusion.}
Therefore, the sum of the first $n$ terms of the AP with first term 1 and common difference 2 is: \[ n^2 \]
Final Answer:} $n^2$.