Problem no. 13

Algebra Level 2

The sum of square of three consecutive natural numbers divided by four is equal to $\frac{151}{2}$ .

Determine the sum of those three natural numbers.

2 solutions

Sravanth C.
Mar 28, 2015

According to the question, $\frac { x^{ 2 }+(x+1)^{ 2 }+(x+2)^{ 2 } }{ 4 } =\frac { 151 }{ 2 }$

Or, ${ x^{ 2 }\quad+\quad x^{ 2 }\quad+\quad2x\quad+\quad1\quad+\quad x^{ 2 }\quad+\quad4x\quad+\quad4 } = { 151 }\times \quad 2$

Or, ${ 3x^{ 2 }\quad +\quad 6x\quad +\quad 5 }={ 151\quad }\times \quad 2$

Or, ${ 3x^{ 2 }\quad +\quad 6x\quad }+\quad 5\quad ={ \quad }302\quad$

Now, by solving the equation or by trial and error we get $x = 9$

Now, the sum of the three numbers is, $x + (x+1) + (x+2)$

Or, $9 + 9 + 1 +9 + 2 = 30$

Instead of trial and error, you could get $3x^{2} + 6x = 297 \Rightarrow x^{2}+2x = 99$

$\Rightarrow x^{2} + 2x + 1 = 100 \Rightarrow (x+1)^{2} = 100 \Rightarrow x+1 = 10 \Rightarrow x = 9$

Feathery Studio - 5 years, 11 months ago

Yeah. That's why I mentioned to solve the equation or by trial and error. $\huge\ddot\smile$

Sravanth C. - 5 years, 11 months ago

Oh, my bad.

Feathery Studio - 5 years, 11 months ago

Ramiel To-ong
May 29, 2015

by direct counting, the numbers are 9,10 and 11