Express the following numbers as the sum of odd numbers using the… - Vidyadip

Question

Express the following numbers as the sum of odd numbers using the pattern given below
(i) $6^3$$\quad$(ii) $8^3$$\quad$(iii) $7^3$
$\begin{array}{r}1=1=1^3 \\ 3+5=8=2^3 \\ 7+9+11=27=3^3 \\ 13+15+17+19=64=4^3 \\ 21+23+25+27+29=125=5^3\end{array}$

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Answer

From the given pattern, we observe that it follows the relation
$\begin{aligned} \text{n}^3=[\text{n}(\text{n}-1)+1]+ {[\text{n}(\text{n}-1)+3] } +[\text{n}(\text{n}-1)+5]+\ldots+\text{n} \text { terms }\end{aligned}$
(i) 216
(ii) 512
(iii) 343

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