NCERT Class X Chapter 5: Arithmetic Progression Example 14(ii)

Question:

Find the sum of : the first n positive integers

Given:

A positive integer n

To Find:

The sum of the first n positive integers: 1 + 2 + 3 + ... + n

Formula:

The sum S of the first n positive integers is given by the formula:

S = n(n + 1) 2

Solution:

Let S be the sum of the first n positive integers.

S = 1 + 2 + 3 + ... + (n - 1) + n

We can also write the sum in reverse order:

S = n + (n - 1) + (n - 2) + ... + 2 + 1

Adding the two equations, we get:

2S = (1 + n) + (2 + n - 1) + (3 + n - 2) + ... + (n - 1 + 2) + (n + 1)

2S = (n + 1) + (n + 1) + (n + 1) + ... + (n + 1) (n times)

Therefore, 2S = n(n + 1)

⇒ S = n(n + 1) 2

Result:

The sum of the first n positive integers is n(n + 1) 2

Next question solution:

NCERT Class X Chapter 5: Arithmetic Progression Example 15

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