# If the sum of the squares of three consecutive integers is 194, what are the numbers?

Jul 15, 2016

$7 , 8 , 9$ or $- 9 , - 8 , - 7$

#### Explanation:

${a}^{2} + {b}^{2} + {c}^{2} = 194$

There are two possible scenarios, either the three numbers go even-odd-even or they go odd-even-odd. Any even number can be written as $2 k , k \in \mathbb{Z}$ and any odd number can be written as $2 j + 1 , j \in Z$. For the first scenario, we have:

${\left(2 k\right)}^{2} + {\left(2 k + 1\right)}^{2} + {\left(2 \left(k + 1\right)\right)}^{2} = 194$

${\left(2 k\right)}^{2} + {\left(2 k + 1\right)}^{2} + {\left(2 k + 2\right)}^{2} = 194$

$4 {k}^{2} + 4 {k}^{2} + 4 k + 1 + 4 {k}^{2} + 8 k + 4 = 194$

$12 {k}^{2} + 12 k + 5 = 194$

We now have a quadratic we can solve for k.

$12 {k}^{2} + 12 k - 189 = 0$

$k = \frac{- 12 \pm \sqrt{144 - 4 \left(12\right) \left(- 189\right)}}{24} = \frac{7}{2} \mathmr{and} - \frac{9}{2}$

These are not integer values so this is not the case.

Trying the second case, we have:

${\left(2 k + 1\right)}^{2} + {\left(2 \left(k + 1\right)\right)}^{2} + {\left(2 \left(k + 1\right) + 1\right)}^{2} = 194$

${\left(2 k + 1\right)}^{2} + {\left(2 k + 2\right)}^{2} + {\left(2 k + 3\right)}^{2} = 194$

$4 {k}^{2} + 4 k + 1 + 4 {k}^{2} + 8 k + 4 + 4 {k}^{2} + 12 k + 9 = 194$

$12 {k}^{2} + 24 k + 14 = 194$

$12 {k}^{2} + 24 k - 180 = 0$

$12 \left({k}^{2} + 2 k - 15\right) = 0$

$12 \ne 0 \implies {k}^{2} + 2 k - 15 = 0$

$\left(k + 5\right) \left(k - 3\right) = 0$

so k is 3 or -5. This means our numbers are:

$7 , 8 , 9$ or $- 9 , - 8 , - 7$

Jul 15, 2016

Three consecutive integers are $\left\{- 9 , - 8 , - 7\right\}$ or $\left\{7 , 8 , 9\right\}$.

#### Explanation:

Let the three numbers be $x - 1$, $x$ and $x + 1$. As sum of their squares is $194$, we have

${\left(x - 1\right)}^{2} + {x}^{2} + {\left(x + 1\right)}^{2} = 194$ or

${x}^{2} - 2 x + 1 + {x}^{2} + {x}^{2} + 2 x + 1 = 194$ or

$3 {x}^{2} + 2 = 194$ or $3 {x}^{2} - 192 = 0$

Or ${x}^{2} - 64 = 0$ i.e. $\left(x + 8\right) \left(x - 8\right) = 0$

Hence $x = - 8$ or $x = 8$ and as this is middle number

Three consecutive integers are $\left\{- 9 , - 8 , - 7\right\}$ or $\left\{7 , 8 , 9\right\}$.