Question

$\text{Sum of}n\text{consecutive odd numbers is divisible by}n.\text{Explain the reason.}$
[3 marks]

Answer 1

$\text{Odd numbers are of the form of}2n-1.\text{where}n=1,2,3,\cdots$
[0.5 mark]
$\text{Now, Sum of}n\text{consecutive odd numbers}\left(S\right)=1+3+5+\cdots +(2n-1)S=(2-1)+(4-1)+(6-1)+\cdots +(2n-1)$
[0.5 mark]
$S=2+4+6+\cdots +2n-\underset{\text{n times}}{\underset{}{(1+1+\cdots +1+1)}}$
[0.5 mark]
$S=2(1+2+3+\cdots +n)-n$
[0.5 mark]
$S=\frac{2\times n\times (n+1)}{2}-nS=n\times (n+1)-nS=n^2$

$\text{Hence, sum of}n\text{consecutive odd numbers is divisible by}n.$
[1 mark]