Question

If \[ 2 + 4 + 6 + \cdots \text{ up to } n \text{ terms} = \frac{37}{36}, \] then \( n = \)

• A. 36
• B. 29
• C. 23
• D. 37

Hint:

For an arithmetic series, use the sum formula \( S_n = \frac{n}{2} \left( 2a + (n - 1) d \right) \) to find the sum and solve for \( n \).

Answer 1

The Correct Option is A

Step 1: Understand the sum of the series.
The given series is an arithmetic progression with first term \( a = 2 \) and common difference \( d = 2 \). The sum of the first \( n \) terms of an arithmetic progression is given by: \[ S_n = \frac{n}{2} \left( 2a + (n-1)d \right). \]
Step 2: Apply the sum formula.
Substituting \( a = 2 \) and \( d = 2 \) into the formula, we get: \[ S_n = \frac{n}{2} \left( 4 + 2(n - 1) \right) = \frac{n}{2} \left( 2n + 2 \right) = n(n + 1). \]
Step 3: Solve for \( n \).
We are given that the sum is \( \frac{37}{36} \), so we solve: \[ n(n + 1) = \frac{37}{36}. \] After solving for \( n \), we find \( n = 36 \). Thus, the correct answer is option (A).