Formula for the Sum of Terms in AP

Q:- Write the Formula for the Sum of Terms in AP.

Answer:- An arithmetic progression is a series or sequence of numbers in which the difference between the successive numbers is the same. There can be infinite numbers in an arithmetic progression. For example, the series of natural numbers, which is 1,2,3,4,5…, is an arithmetic progression that has the same difference between the successive numbers. 

The Sum of Terms in AP

The sum of the n terms in an arithmetic progression can be easily found by using a simple formula which states that if we have an arithmetic progression who’s the first term is a and the common difference between the terms is d, then the formula for the n term of AP is

Sₙ = n/2[2a+(n-1)d]

Notation in Arithmetic Progression 

In an arithmetic progression, the term which we come across frequently are:-

• First-term (a)

• Common difference (d)

• nth term (an)

• Sum of the first n term (Sn)

First-Term in AP 

An AP can also be written in common differences for example:- 

a, a+d, a+2d, a+3d……..a + (n-1)d 

Where a is the first term in an arithmetic progression. 

The Common Difference in AP

In an AP for a series, the common difference is the difference between the two successive terms; it can be obtained by using the formula 

d=a2-a1 = a3-a2……=an – an-1 

Where d is a common difference, it can be positive, negative or zero. 

Sum of n terms in AP when the last term is given.

The sum of the n terms of an AP when the last term is known is:-

Sₙ=n/2×[a1+an]

Sum of AP Formula for an Infinite AP

Let’s take an example of the sum of an infinite AP

2+5+8…

Here,

a=2 

d=3 

The number of terms n=∞

Substituting the values in the AP formula 

Sₙ=n/2 (2a+(n-1)d)

Sₙ=∞/2(2(2)+(∞-1)3)

Sₙ=∞

The sum of infinite AP is ∞ when d>0. And the sum of infinite AP will be -∞ when d<0.