Question

The sum of the first 10 terms of an AP with \(a=2\) and \(d=3\) is: 

• A. 155
• B. 145
• C. 165
• D. 150 Correct Answer: (A) 155 Solution:

Hint:

For AP problems involving sums, memorize: \[ S_n=\frac{n}{2}\left[2a+(n-1)d\right] \] It is one of the most frequently used formulas in sequences and series.

Answer 1

The Correct Option is A

Concept: The sum of the first \(n\) terms of an arithmetic progression (A.P.) is given by \[ S_n=\frac{n}{2}\left[2a+(n-1)d\right] \] where

• \(a\) = first term

• \(d\) = common difference

• \(n\) = number of terms

Step 1: Write the given values. \[ a=2,\qquad d=3,\qquad n=10 \]

Step 2: Substitute into the sum formula. \[ S_{10} = \frac{10}{2} \left[ 2(2)+(10-1)(3) \right] \] \[ = 5 \left[ 4+27 \right] \] \[ = 5(31) \] \[ = 155 \]

Step 3: Verify by listing the terms. The first ten terms are \[ 2,\;5,\;8,\;11,\;14,\;17,\;20,\;23,\;26,\;29 \] Adding them also gives \[ 155 \] Therefore, \[ \boxed{S_{10}=155} \] Hence, the correct option is \(\boxed{(A)}\).