Question

Express 100 as the sum of 10 odd natural number

Answer 1

Using Arithmetic Progression, we are going to express it.

Formula used: Sum of n odd numbers (odd or even or natural or whole, the answer is based on the difference between 2 successive terms in the series)
                              Sₓ = 1/2 x n[2a+(n-1)×d]

Series be 1,3,5,7,9,11... (odd numbers)
Total numbers n = 10
First number in the sequence a = 1
Difference between two successive terms d = 3-1 = 2

Sum of n odd numbers: Sₓ = 1/2 x n[2a+(n-1)×d]
                                      S₁₀ = 1/2 x 10[2(1)+(10-1)2]
                                      S₁₀ = 1/2 x 10[2+18]
                                      S₁₀ = 5(20)
                                      S₁₀ = 100

Hence, It is proved.

Answer 2

Hi,

i ) 1 = 1²

ii ) 1 + 3 = 4 = 2²

iii ) 1 + 3 + 5 = 9 = 3²

iv ) 1+ 3+5+7 = 16 = 4²

v ) 1+3+5+7+9 = 25 = 5²

Here observe the pattern,

Sum of ' n ' consecutive

odd numbers = n²

Sum of 10 consecutive

odd numbers = (10)² =100

Therefore,

( 1+3+5+7+9+11+13+15+17+

19) = 10² = 100

I hope this helps you.

:)