Question

Find the value of S where,

S = 1² - 2² + 3² - 4² + 5² - 6² +....100²

Show your working. ​

Answer 1

Step-by-step explanation:

this question's answer is -5050

Answer 2

The sum of the series

$S=1^2-2^2+3^2-4^2+5^2-6^2+...100^2$

can be found as the following method. [1]

Regrouping each 2nd term, [2]

$S=(1^2-2^2)+(3^2-4^2)+(5^2-6^2)+(99^2-100^2)$

$\Longleftrightarrow S=(1+2)(1-2)+(3+4)(3-4)+(5+6)(5-6)+...+(99+100)(99-100)$

$\Longleftrightarrow S=-(1+2)-(3+4)-(5+6)-...-(99+100)$

$\Longleftrightarrow S=-(1+2+3+4+5+6+...+99+100)$

Rearranging,

• $-S=1+2+3+...+98+99+100$
• $-S=100+99+98+...+3+2+1$

Add the two equations to get

$-2S=101+101+101+...+101+101+101$.

Note there are one hundred 101s in the series.

Then,

$-2S=10100$

$\therefore S=-5050$

Hence, the value of the sum is -5050.

More information

[1] Gauss's method is used.

[2] Geometric approach can be used when calculating the difference. Refer to the attachment.