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

mid term is \(T _{26}=300\)
\(T _{1}=300-25 d ; T _{51}=300+25 d\)
\(S =\frac{51}{2}[300-25 d +300+25 d ]\)
\(\frac{51}{2}[600]=15,300\)

 To find the sum of the first 51 terms of an arithmetic progression (A.P.), we need to use the formula for the sum of an A.P., which is given by:

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

where: Sn is the sum of the first n terms, a is the first term, n is the number of terms, and d is the common difference.

In this case, we are given that the middle term of the A.P. is 300. Let's assume the middle term is also the 26th term (n = 26).

We know that the middle term of an arithmetic progression is given by:

a + (n-1)d/2

Substituting the values:

300 = a + (26-1)d/2 600 = 2a + 25d

We have two equations:

1. 600 = 2a + 25d
2. 300 = a + 25d

Solving these two equations simultaneously, we can find the values of a and d.

Subtracting equation 2 from equation 1, we get: 300 = a

Substituting this value back into equation 2: 300 = 300 + 25d 25d = 0 d = 0

We have found that the common difference (d) is 0. This means that all the terms in the arithmetic progression are the same, and the sum of the first 51 terms is simply 51 times the middle term.

Sum = 51 * 300 = 15,300

Therefore, the sum of the first 51 terms is 15,300. So the correct answer is 15300.