Sum of N Terms of an AP

The sum of n terms of an Arithmetic Progression (AP) is the total obtained by adding the first n terms of the sequence. If the first term is a, the common difference is d, and the number of terms is n, then:

Sum of n Terms of AP (When Last Term is Given)

The sum of the first n terms of an Arithmetic Progression (AP), when the first term (a₁) and the last term (aₙ) are known, is:

Sn = n/2 × (a₁ + aₙ)

Example: Sum of Numbers from 1 to 100

To find the sum from 1 to 100, we pair terms from the beginning and the end:

(1, 100), (2, 99), (3, 98), …

Each pair = 101
Number of pairs = 50

So,
Sum = 101 × 50 = 5050

Sum of n Terms of AP Proof

Let us consider a general Arithmetic Progression (AP). The sum of its first n terms is given by:

Sn = a + (a + d) + … + (a + (n − 1)d) … (1)

Sn = (a + (n − 1)d) + … + (a + d) + a … (2)

//image due-

Adding (1) and (2):

2Sn = n [2a + (n − 1)d]

Sn = n/2 [2a + (n − 1)d]

Also, since an = a + (n − 1)d,

Sn = n/2 (a + an)

Thus, the sum of n terms of the AP formula is Proved.

Solved Examples

Example 1: Find the sum of squares of the first 10 natural numbers.

Solution:

We know that formula for the sum of n terms of the square of the natural number is,

S_n = n(n + 1)(2n + 1)/6

For finding the square of the sum of the first 10 natural numbers
S_n = 10(10 + 1)(20 + 1)6
= 10(11)(21)/6
= 385

Thus, the sum of the first 10 squares of the natural number is 385.

Example 2: Consider the AP = 2, 4, 6, 8, 10,... Find the sum of the first 20 terms of this A.P.

Solution:

Given AP: 2, 4, 6, 8, 10,...
First Term (a) = 2
Common Difference (d) = 4 - 2 = 6 - 4 = 2

Sum of n terms is,
S_n = n/2 [2a + (n - 1)d]
n = 20

S_n = 20/2 [2 × 2 + (20 - 1) × 2]
S_n = 10(42) = 420

Thus, the sum of the of first 20 terms in the sequence are, 420

Example 3: Calculate the sum of the first 16 terms of the AP: S = 98 + 95 + 92 +...

Solution:

Given AP: S = 98 + 95 + 92 +...
First Term (a) = 98
Common Difference (d) = 95 - 98 = 92 - 95 = -3

Sum of n terms is,
S_n = n/2 [2a + (n - 1)d]
n = 16
S_n = 16/2 [2 × 98 + (16 - 1) × (-3)]
S_n = 8(151) = 1208

Thus, the sum of the of first 16 terms in the sequence are, 1208

Example 4: Calculate the sum of the first 24 terms of the AP: S = 8 + 13 + 18 +...

Solution:

Given AP: S = 8 + 13 + 18 +...

First Term (a) = 8
Common Difference (d) = 13 - 8 = 18 - 13 = 5

Sum of n terms is,
S_n = n/2 [2a + (n - 1)d]
n = 24
S_n = 24/2 [2 × 8 + (24 - 1) × 5]
S_n = 12(131) = 1572

Thus, the sum of the of first 24 terms in the sequence are, 1572