Approximating an area by rectangles

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SUMMARY

The discussion focuses on approximating areas using rectangles by summing identities for series. Specifically, it addresses finding the sum of the first n natural numbers using the identity (m + 1)² - m² = 2m + 1, and the sum of the squares of the first n natural numbers using (m + 1)³ - m³ = 3m² + 3m + 1. The user bluemoon2188 suggests deriving recursive formulas for these sums based on the provided identities, facilitating a deeper understanding of series summation.

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bluemoon2188
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Hi,

I have this problem,

1) Find 1 + 2 + · · · + n by summing the identity (m + 1)2 − m2 = 2m + 1 from m = 1 to n.
2) Similarly find 12 + 22 + · · · + n2 using the identity (m + 1)3 − m3 = 3m2 + 3m + 1

Thanks in advance.
bluemoon2188
 
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Consider the summing the identities provided from 1 through to n. You should be able to obtain a recursive formula for each power in terms of the sums of the lower powers.
 
oh cool thanks.

bluemoon2188
 

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