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Mathematics · Arithmetic Progressions · NCERT Examples

Example 14

A complete, independently verified solution for NCERT Example 14.

Mathematics · Chapter 5 · NCERT Examples

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NCERT Example

Example 14

sum of AP

Find (i) the sum of the first 1000 positive integers and (ii) the sum of the first n positive integers.

Solution

Board-exam working:

Sn=n2[2a+(n−1)d]=n2(a+l)S_n=\frac{n}{2}[2a+(n-1)d]=\frac{n}{2}(a+l)

The positive integers form an AP with a=1,d=1.

Use Sₙ=n(a+l)/2.

(i) Here n=1000 and l=1000.

S₁₀₀₀=1000/2(1+1000)=500×1001=500500.

(ii) For the first n positive integers, l=n.

Therefore Sₙ=n/2(1+n)=n(n+1)/2.

Final answer

(i) 500500; (ii) n(n+1)/2.