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# Express the following as the sum of two consecutive integers1) ${21^2}$2) ${13^2}$3) ${11^2}$4) ${19^2}$

Last updated date: 20th Jun 2024
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Hint: Here we will express the given number in two consecutive integers. Firstly we will let one integer and take another integer as its successive number. Then we will add the two consecutive term integers we have left and put it equal to the number. Finally we will get our desired answer.

We will let one integer as $x$ in each case and since the integers are consecutive we will let another integer as $x + 1$.
1. Let us take two integers as $x$ and $x + 1$.
So, as we have to put the sum of above two integers equal to the number we get,
$\left( x \right) + \left( {x + 1} \right) = {21^2} \\ \Rightarrow 2x + 1 = 441 \\ \Rightarrow 2x = 441 - 1 \\ \Rightarrow x = \dfrac{{440}}{2} = 220 \\$
So, $x + 1 = 220 + 1 = 221$
So we get our integers as $220$and $221$
2. Let us take two integers as $x$ and $x + 1$.
So, as we have to put the sum of above two integers equal to the number we get,
$\left( x \right) + \left( {x + 1} \right) = {13^2} \\ \Rightarrow 2x + 1 = 169 \\ \Rightarrow 2x = 169 - 1 \\ \Rightarrow x = \dfrac{{168}}{2} = 84 \\$
So, $x + 1 = 84 + 1 = 85$
So we get our integers as $84$and $85$
3. Let us take two integers as $x$ and $x + 1$.
So, as we have to put the sum of above two integers equal to the number we get,
$\left( x \right) + \left( {x + 1} \right) = {11^2} \\ \Rightarrow 2x + 1 = 121 \\ \Rightarrow 2x = 121 - 1 \\ \Rightarrow x = \dfrac{{120}}{2} = 60 \\$
So, $x + 1 = 60 + 1 = 61$
So we get our integers as $60$and $61$
4. Let us take two integers as $x$and $x + 1$.
So, as we have to put the sum of above two integers equal to the number we get,
$\left( x \right) + \left( {x + 1} \right) = {19^2} \\ \Rightarrow 2x + 1 = 361 \\ \Rightarrow 2x = 361 - 1 \\ \Rightarrow x = \dfrac{{360}}{2} = 180 \\$
So, $x + 1 = 180 + 1 = 181$

So we get our integers as $180$ and $181$

Note:
Consecutive integers are those which come one after the other. If a limitation is given such that the number should be an even consecutive number or odd consecutive number then they are written as$... - 4, - 2,0,2,4...$,$... - 5, - 3, - 1,1,3,5....$ respectively. The formula for consecutive numbers is given as $x + 1,x + 2....$ where the second number can go up to infinity.