Question

The sum of the squares of four
consecutive natural numbers is 294.
What is the sum of the numbers ?​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{Sum of the squares of four consecutive}$

$\textsf{natural numbers is 294}$

$\underline{\textbf{To find:}}$

$\textsf{Sum of the four numbers}$

$\underline{\textbf{Solution:}}$

$\textsf{Let the consecutive of four natural numbers be}$

$\textsf{n,n+1,n+2 and n+3}$

$\textsf{As per given data,}$

$\mathsf{n^2+(n+1)^2+(n+2)^2+(n+3)^2=294}$

$\mathsf{n^2+(n^2+1+2n)+(n^2+4+4n)+(n^2+9+6n)=294}$

$\mathsf{4n^2+12n+14=294}$

$\mathsf{4n^2+12n-280=0}$

$\textsf{Divide by 4 on bothsides of the equation}$

$\mathsf{n^2+3n-70=0}$

$\mathsf{n^2+10n-7n-70=0}$

$\mathsf{n(n+10)-7(n+10)=0}$

$\mathsf{(n+10)(n-7)=0}$

$\mathsf{n=7,-10}$

$\textsf{Since n is a natural number, n can't be negative}$

$\implies\mathsf{n=7}$

$\textbf{The four numbers are 7,8,9 and 10}$

$\textbf{Sum of the numbers=7+8+9+10=34}$