Find the value of S where,
S = 1² - 2² + 3² - 4² + 5² - 6² +....100²
Show your working.
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Find the value of S where,
S = 1² - 2² + 3² - 4² + 5² - 6² +....100²
Show your working.
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Verified answer
Step-by-step explanation:
this question's answer is -5050
The sum of the series
[tex]S=1^2-2^2+3^2-4^2+5^2-6^2+...100^2[/tex]
can be found as the following method. [1]
Regrouping each 2nd term, [2]
[tex]S=(1^2-2^2)+(3^2-4^2)+(5^2-6^2)+(99^2-100^2)[/tex]
[tex]\Longleftrightarrow S=(1+2)(1-2)+(3+4)(3-4)+(5+6)(5-6)+...+(99+100)(99-100)[/tex]
[tex]\Longleftrightarrow S=-(1+2)-(3+4)-(5+6)-...-(99+100)[/tex]
[tex]\Longleftrightarrow S=-(1+2+3+4+5+6+...+99+100)[/tex]
Rearranging,
Add the two equations to get
[tex]-2S=101+101+101+...+101+101+101[/tex].
Note there are one hundred 101s in the series.
Then,
[tex]-2S=10100[/tex]
[tex]\therefore S=-5050[/tex]
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.