CBSE Class 10 Mathematics chapter 5 - ARITHMETIC PROGRESSIONS - Sum of First n Terms of an AP

 CBSE Class 10 Mathematics chapter 5 - ARITHMETIC PROGRESSIONS - Sum of First n Terms of an AP

Sum of First n Terms of an AP:

We know that n^th term of AP = a + (n – 1) d

Let S represent the sum of the first n terms of the AP. So we can write –

S = a + (a + d ) + (a + 2d) + . . . + [a + (n – 1) d ] …………….. (i)

When we rewrite this terms in reverse order, we have-

S = [a + (n – 1) d] + [a + (n – 2) d ] + . . . + (a + d) + a ……………. (ii)

By adding term (i) and (ii) term-wise we get

2S = n [2a + (n – 1) d ]             (since, there are n terms)

So, the sum of the first n terms of an AP is

Note : these terms are used when only first and common difference of AP are given.

We know that a_n = a + (n – 1) d

so we can write as

If an AP has only n terms then a_n = l

Note : these term are used when only first and last terms of AP are given.

Note : The n^th term of an AP is the difference of the sum to first n terms and the sum to first (n – 1) terms of it.

a_n = S_n – S_{n – 1}

Note: If ‘a’ is positive and ‘d’ is negative, then some terms will be positive and some others will be negative and will cancel out each other.

Note: The sum of first ‘n’ positive  integers given by this formula-