[Series] Sum of Odd Numbers

2021-07-06 11:57 作者:AoiSTZ23 0人讀過 | 我要投稿

By: Tao Steven Zheng (鄭濤)

【Problem】

The following problem is attributed to Pythagoras of Samos (c. 570 - 495 BC) and his Pythagorean sect.

Determine the formula for calculating sum of consecutive odd numbers:

$S_n%3D1%2B3%2B5%2B%E2%8B%AF%2B(2n-1)$

?

【Solution】

Pythagoras solved this problem geometrically. Figure 1 visually demonstates that the sum of consecutive odd numbers (starting from 1) is a perfect square.

This problem is an arithmetic series. The first term is $a_1%3D1$ ?, and the final term is $%20a_n%3D2n-1$ .

Hence,

$S_n%3D%5Cfrac%7Bn%7D%7B2%7D%20(a_1%2Ba_n%20)%0A%0A$

$S_n%3D%5Cfrac%7Bn%7D%7B2%7D%20(1%2B2n-1)%0A$

$S_n%3Dn%5E2%0A$

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