Question

If the middle term of the A.P is 300 then the sum of its first 51 terms is

• A. 15300
• B. 14800
• C. 16500
• D. 14300

Answer 1

The Correct Option is A

The sum of the first \( n \) terms of an arithmetic progression (A.P.) is given by the formula: \[ S_n = \frac{n}{2} \times \left(2a + (n-1)d \right) \] where:
\( a \) is the first term,
\( d \) is the common difference,
\( n \) is the number of terms.
In this case, the number of terms is 51, and the middle term is given as 300. Since the middle term of an A.P. is the \( \left(\frac{51+1}{2}\right) = 26 \)-th term, we have: \[ a + (26-1)d = 300 \] Simplifying, we get: \[ a + 25d = 300 \] Now, using the formula for the sum of the first 51 terms: \[ S_{51} = \frac{51}{2} \times \left(2a + (51-1)d \right) \] \[ S_{51} = \frac{51}{2} \times \left(2a + 50d \right) \] Substituting \( a + 25d = 300 \) into the equation: \[ S_{51} = 51 \times 300 = 15300 \] Therefore, the sum of the first 51 terms is \( 15300 \).