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

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Solution

(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:
$43 + 45 + 47 + 49 + 51 + 53 + 55$

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