If the 10th term of an Ap is -27 and the 5th term is -12

Question

What will be the sum of its 25 terms ?

Answer 1

First, we need to find the common difference, denoted by 'd', of the arithmetic progression (AP).

We are given that the 10th term is -27 and the 5th term is -12.

To find the common difference, we can use the formula for the nth term of an AP:

an = a1 + (n-1)d

where an is the nth term, a1 is the first term, and d is the common difference.

Using the information given, we have:

-27 = a1 + (10-1)d ... (1)

-12 = a1 + (5-1)d ... (2)

Solve equations (1) and (2) simultaneously.

From equation (2), rearrange to express a1 in terms of d:

a1 = -12 - 4d ... (3)

Substitute equation (3) into equation (1):

-27 = (-12 - 4d) + (10-1)d

-27 = -12 - 4d + 10d - d

-27 = -12 + 6d

Rearrange the equation:

6d = -27 + 12

6d = -15

d = -15/6

Simplify d:

d = -5/2

Now, we know that d = -5/2.

To find the sum of 25 terms, we will use the formula for the sum of an AP:

Sn = (n/2)(2a1 + (n-1)d)

where Sn is the sum of the first n terms.

Substituting in the given values, we have:

S25 = (25/2)(2a1 + (25-1)d)

Simplify:

S25 = (25/2)(2(-12 - 4d) + (25-1)d)

Simplify further by substituting the value of d:

S25 = (25/2)(2(-12 - 4(-5/2)) + (25-1)(-5/2))

Simplify and calculate:

S25 = (25/2)(2(-12 + 10) - (24)(5/2))

Simplify further:

S25 = (25/2)(2(-2) - 60/2)

S25 = (25/2)(-4 - 30)

S25 = (25/2)(-34)

Multiply:

S25 = -425

Therefore, the sum of the 25 terms of the AP is -425.