# Arithmetic and Geometric Progressions

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$n(n + 1)^{2} = n^{3} + 2n^{2} + n$

## Sum of a Sequence

Find sum : $2^{2} + 2 * 3^{2} + 3 * 4^{2} + 4 * 5^{2}.....10 * 11^{2}$
1. 6530
2. 3600
3. 2850
4. 3850

Choice D. 3850

## Detailed Solution

The series is in the form $n(n + 1)^{2} = n^{3} + 2n^{2} + n$

Then $\Sigma n^{3} + 2n^{2} + n = [\frac{n(n + 1)}{2} ]^{2} + 2 [n (n + 1)\frac{2n + 1)}{6}] + [\frac{n(n + 1)}{2}]$
Substituting n = 10 will give the sum of the series. Thus $\Sigma [\frac{10 * 11)}{2}]^{2} + [\frac{10 * 11 * 21)}{6} ] + [\frac{10 * 11)}{2}]$

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## More questions from Progressions

With some simple but very powerful ideas, one can cut down on a lot of working when it comes to progressions. For example, anchoring a progression around its middle term can be very useful. Reinforce these ideas with these questions.