The sum of the first three terms of the arithmetic progression is 30 and the sum of the squares of the first term and the second term of the same progression is 116. Find the seventh term of the progression if its fith term is known to be exactly divisible by 14.

Options

A. 36
B. 40
C. 43
D. 22

Answer

To solve the problem step by step, we will denote the first term of the arithmetic progression (AP) as \( a \) and the common difference as \( d \).

### Step 1: Set up the equations based on the problem statement.
1. The sum of the first three terms of the AP is given as 30:
\[ (a - d) + a + (a + d) = 30 \]
Simplifying this:
\[ 3a = 30 \implies a = 10 \]