Problem.

The sum of certain consecutive odd numbers is $\displaystyle 57^2$–$\displaystyle 13^2.$ Find the numbers.

solution

$\displaystyle \sum 1+3+5+....+(2n-1) = n^2 [1] $

$\displaystyle 57^2=n^2$

$\displaystyle n=57$

also from equ 1

$\displaystyle n=\frac{l+1}{2}$

therefore

$\displaystyle \frac{l_1+1}{2} = 57$

$\displaystyle l_1=113$

similarly

$\displaystyle 13^2=n^2$

$\displaystyle l_2=25$

$\displaystyle 57^2 - 13 ^2 = 27^2+29^2+....113^2$

a = 27 , d = 2

l= 113

a+nd = 113

27 + 2n = 113

n = 43

this is where I am facing the problem. To satisfy the condition I should have n = 44 and not 43.

because 57 - 13 = 44. other way to find the n.

Please correct my understanding...