Sum of first 4 terms of arithmetic progression is 28 and sum of first eight terms is 88. Find sum of first 10 terms.

Question

Sum of first 4 terms of arithmetic progression is 28 and sum of first eight terms is 88. Find sum of first 10 terms.

Answer 1

The formula for the sum of the first n terms of an arithmetic progression is (2a + d(n-1))n / 2, where

 

a is the first term

d is the difference between terms.

 

(I'm using a for the first term instead of a₁ to make it easier to type.)

 

So by setting n = 4, we know from the first statement:

 

(2a + d(4-1))4 / 2 = 28

(2a + d(3))4 / 2 =  28

2 (2a + 3d) = 28

2a + 3d = 14

 

With n = 8, we know from the second statement:

 

(2a + d(8-1))8 / 2 = 88

(2a + 7d)4 = 88

2a + 7d = 22

 

So we have two equations in two unknowns that we can solve:

 

2a + 3d = 14

2a + 7d = 22

 

4d = 8

d = 2

 

Substitute d back into the first equation:

2a + 3*2 = 14

2a + 6 = 14

2a = 8

a = 4

 

So that means that the first term in the progression is 4 and the difference between terms is 2.

 

To confirm this, the first 4 terms are 4, 6, 8, 10 -- and their sum is 28 as expected.

 

And the first 8 terms are 4, 6, 8, 10, 12, 14, 16, 18 -- and their sum is 88!

 

So now you can use what you know about the progression to find the sum of the first 10 terms.

 

Either plug the values for a and d into the formula: 

 

(2*4 + d(n-1))n / 2 =

(8 + 2(10-1))10 / 2 =

(8 + 2*9)5 =

26 * 5 =

130

 

Or start with the first 8 terms -- 4, 6, 8, 10, 12, 14, 16, 18 -- for which you already know the sum is 88, and add on two more terms -- 20, 22 -- and see what happens.

 

20 + 22 = 42

88 + 42 = 130

 

So the answer is 130

 

Answer 2

Hi Neha,

 

sum of n terms of AP = S_n= n/2 *[ 2a +(n-1)d]

 

S₄ = 4/2 [2a +(4-1)d]

28 = 2(2a +3d)

 

14 = 2a +3d........Eq 1

 

S₈ = 8/2 [2a +(8-1)d]

 

88 = 4(2a +7d)

 

22 = 2a +7d........ Eq 2

 

subtract Eq 1 from Eq 2, we get

 

8=4d

2 = d

 

substitute d=2 in Eq 1,

 

14 =2a +6

 

8 = 2a

a= 4

 

S₁₀ = 10/2[8 +18]

       =5*26

       =130

Ans: sum of 10 terms=130