Formula to find sum of n terms in ap

Asked by admin @ 20/08/2021 in Math viewed by 330 People

What is formula to find the sum of n terms of AP

Answered by admin @ 20/08/2021

Answer:The formula says that the sum of the first n terms of our arithmetic sequence is equal to n divided by 2 times the sum of twice the beginning term, a, and the product of d, the common difference, and n minus 1. The n stands for the number of terms we are adding together.

Sum of arithmetic progression

The sum of arithmetic progression is denoted by Sn. It is nothing but the sum of 'n terms of an A.P. with first term 'a' and common difference 'd'.

The formula for sum of n terms of A.P. is

Sn=

[2a+(n−1)d]

Sn=

[a+l] , where l = last term = a + (n -1 )d

Proof : Let a1,a2,a3,...,an be an A.P. with first term as 'a' and common difference as 'd'.

a1 = a; a2 = a + d; a3 = a + 2d ; ... an = a + (n -1)d

Sn=a1+a2+a3+...+an−1+an

⇒ Sn = a + (a + d) + (a + 2d) + … + [ a + (n-2)d] + [ a + (n -1 )d] -----(i)

Write the above equation in reverse order we get,

Sn = [ a + (n -1 )d] + [ a + (n-2)d] + … + (a + 2d) + (a + d) + a ----- (ii)

Now add the two equations,

2Sn = [ 2a + (n -1 )d] + [ 2a + (n-1)d] + …+ [2a + (n-1)d]

[2a + (n-1)d] repeats ‘n’ times

∴ 2Sn = n [ 2a + (n -1 )d]

Sn=

[ 2a + (n -1 )d]

Since the last term l = a + (n – 1)d

∴ Sn=

[ a + a + (n -1 )d]

Sn=

[a + l ]

Note : In the above formula there are 4 unknown quantities. So if any three are given then we can find the forth one.

In the sum of Sn of n terms of a sequence is given then the nth term an of the sequence can determined by using the following formula .

Formula to find sum of n terms in ap

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