Summation problem

Evaluate $\displaystyle{\sum_{a= 3}^{6} (a-2)^2}$ and $\displaystyle{\sum_{a=1}^{4} a^2}$ .

     What do you notice?
     Why does this work?

#Algebra

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Well, the value of both of them is same, i.e., $30$ .

$\displaystyle \sum^{6}_{a=3}(a-2)^2 \\ \text{Let, } n=a-2\Rightarrow a=n+2 \\ \displaystyle \sum^{n+2=6}_{n+2=3}n^2=\displaystyle \sum^{n=4}_{n=1}n^2 \text{ or } \displaystyle \sum^{4}_{n=1}n^2$

Akshat Sharda - 5 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$