Question

What is the sum of the first 10 terms of the arithmetic progression with first term 3 and common difference 2?

• A. 120
• B. 105
• C. 75
• D. 90

Hint:

The sum of an A.P. can be calculated using the first term, common difference, and number of terms, or by averaging the first and last terms multiplied by the number of terms.

Answer 1

The Correct Option is A

Step 1: Identify the formula for the sum of the first \( n \) terms of an arithmetic progression (A.P.): \[ S_n = \frac{n}{2} [2a + (n - 1)d], \] where \( n \) is the number of terms, \( a \) is the first term, and \( d \) is the common difference.
Step 2: Substitute the given values:
- \( n = 10 \),
- \( a = 3 \),
- \( d = 2 \).
\[ S_{10} = \frac{10}{2} [2 \cdot 3 + (10 - 1) \cdot 2]. \]
Step 3: Perform the calculation step-by-step:
- Compute the number of terms and factor: \[ S_{10} = 5 [6 + 9 \cdot 2]. \] - Calculate the expression inside the brackets: \[ 9 \cdot 2 = 18, \quad 6 + 18 = 24. \] - Multiply by 5: \[ S_{10} = 5 \cdot 24 = 120. \]
Step 4: Verify the result.
The sequence is 3, 5, 7, ..., up to the 10th term.
The 10th term is \( a + (n - 1)d = 3 + 9 \cdot 2 = 21 \), and the sum of an A.P. can also be \( S_n = \frac{n}{2} (a + l) \), where \( l \) is the last term: \[ S_{10} = \frac{10}{2} (3 + 21) = 5 \cdot 24 = 120. \]