Question

Find the sum of the first 51 terms of an A.P whose middle term is 300.

Answer 1

middle term of the ap will be 26..
n=26, an= a+(n-1)d= a26=a+25d=300
S51=51/2[2a+50d] = 51/2[2(a+25d) =51/2[2*300] = 51*300 = 15300

Answer 2

Given:

Middle term = 300

To find:

The sum of the first 51 terms.

Solution:

We know that the formula for the sum of n terms of an A.P,

Sn = n/2 { 2 a + ( n- 1) d }

n is the total number of terms in the A.P;

a = first term

d= common difference

Here, Sn= S₅₁

S₅₁ = 51/2  { 2 a + ( 51 - 1) d }

S₅₁ = 51/2  { 2 a + 50 d }

S₅₁ = 51/2  { 2 ( a + 25 d ) }                                             ( Equation 1 )

The middle term will be the 26th term in the A.P

a₂₆ = a + ( 26 -1 ) d = 300

a + 25 d = 300                                                                ( Equation 2 )

Putting the value of equation 2 in equation 1,

S₅₁ = 51/2  { 2 × 300 }  

S₅₁ = 30,600 ÷ 2

S₅₁ = 15,300

The sum of the first 51 terms of an arithmetic progression is 15300.