Question

Express the following numbers as the sum of consecutive odd numbers:
(i) $8^3$ (ii) $7^3$

Answer 1

(i) $8^3$
Here, the sum of 8 consecutive odd numbers should be equal to $8^3$.
To find the odd numbers whose sum is $8^3$, we first divide $8^3$ by 8.
$\Rightarrow \frac{8^3}{8}=8^2=64$
Since 64 is an even number, there would be two middle terms 63 and 65 in the series. The other odd numbers would be the 3 odd ones before 63 and after 65.
So, $8^3$ can be expressed as:
$57+59+61+63+65+67+69+71$

(ii) $7^3$
Here, the sum of 7 consecutive odd numbers should be equal to $7^3$.
To find the odd numbers whose sum is $7^3$, we first divide $7^3$ by 7.
$\Rightarrow \frac{7^3}{7}=7^2=49$
Since 49 is an odd number, it would be the middle number of the series.
The other odd numbers would be 3 odd numbers before and after 49.
Thus, $7^3$ can be expressed as:
$43+45+47+49+51+53+55$