Question

Express $81$ as the sum of $9$ odd numbers.

Answer 1

Hint: Here we have been given a number which we have to express as the sum of $9$ odd numbers. Firstly we will write the formula for the sum of the first $n$ odd numbers then we will convert the number given to us so that we can compare it by the formula and get the value of $n$ . Finally we will write the number given as the sum of the numbers we obtained and get our desired answer.

Complete step-by-step solution:
We have to express the below number as a sum of $9$ odd numbers,
$81$…..$\left( 1 \right)$
Now as we know that according to the property of perfect square for any $n$ natural number the sum of first $n$ odd natural numbers is as follows,
Sum of $n$ odd numbers $={{n}^{2}}$….$\left( 2 \right)$
We can rewrite the equation (1) as a perfect square as follows,
$\Rightarrow 81={{\left( 9 \right)}^{2}}$
Comparing the above value with equation (2) we get,
$\Rightarrow {{n}^{2}}={{\left( 9 \right)}^{2}}$
On comparing we get,
$n=9$
Which means $81$ can be written as sum of first $9$ natural numbers which are,
$1,3,5,7,9,11,13,15,17$
So we get,
$81=1+3+5+7+9+11+13+15+17$
Hence $81$ is expressed as a sum of $9$ odd numbers as $81=1+3+5+7+9+11+13+15+17$ .

Note: Natural numbers are a part of the number system which don’t include the negative numbers or zero but include all positive integers from $1$ to infinity. In the number line all the integers on the right-hand side of zero represent natural numbers. It is denoted by $N$ and satisfies four main properties which are Closure property, Commutative property, Associative property and distributive property. Odd numbers are those integers that are not divided by $2$ completely which means it leaves a remainder when divided by $2$ .