Arithmetic Sequences

homeschool girl

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Feb 6, 2020
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part (a) Compute the sum \(\displaystyle 101^2 - 97^2 + 93^2 - 89^2 + \cdots + 5^2 - 1^2.\)

part (b) Compute the sum \(\displaystyle (a +(2n+1)d)^2- (a + (2n)d)^2 +(a + (2n-1)d)^2 - (a+(2n-2)d)^2 + \cdots + (a+d)^2 - a^2.\)

I know I need to pair the terms together and apply difference of squares, but don't know which terms I should pair and what to do after. I would very much appreciate some help. thanks!
 
Last edited:

Dr.Peterson

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What you need to do is to try something and see what happens! Don't wait for someone to tell you what to do; figure it out by trying.

You have an idea (given to you as a hint?), so just take a pair of adjacent terms and apply that idea. Then show us what you find. If that doesn't lead in a good direction, try something else, such as the first term and the last term. (I suspect several approaches will work.)
 

homeschool girl

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5304 for part a?
 
Last edited:

homeschool girl

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my work





\(\displaystyle 101-4\cdot (n-1)=1\)



\(\displaystyle 101-4n+4=1\)



\(\displaystyle 105-1=4n\)



\(\displaystyle n=26\)







\(\displaystyle S=101^2 - 97^2 + 93^2 - 89^2 + \cdots+13^2 -9^2 + 5^2 - 1^2.\)



\(\displaystyle S=101^2 - 97^2 + 93^2 - 89^2 + \cdots+13^2 -9^2 + 5^2 - 1^2\)



\(\displaystyle S=(101 - 97)(101 + 97) + (93 - 89)(93 + 89) + \cdots+(13 -9)(13 +9) + (5 - 1)(5 + 1)\)



\(\displaystyle S=792+728 + \cdots+24+88.\)





\(\displaystyle S=792+728 + \cdots+24+88\)

\(\displaystyle +\)

\(\displaystyle S=24+88+\cdots+792+728.\)





\(\displaystyle 2S=816+816+\cdots+816+816.\)



\(\displaystyle 2S=816\cdot13\)



\(\displaystyle S=\frac{816\cdot13}{2}.\)



\(\displaystyle S=5304\)



\(\displaystyle 101^2 - 97^2 + 93^2 - 89^2 + \cdots + 5^2 - 1^2 = \boxed{5304}\)
 

pka

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Well done homeschool-girl, well done.

I had \(\sum\limits_{n = 0}^{25} {{{( - 1)}^n}{{(101 - 4n)}^2}} \) which is the same as yours. See Here
 

Subhotosh Khan

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my work





\(\displaystyle 101-4\cdot (n-1)=1\)



\(\displaystyle 101-4n+4=1\)



\(\displaystyle 105-1=4n\)



\(\displaystyle n=26\)







\(\displaystyle S=101^2 - 97^2 + 93^2 - 89^2 + \cdots+13^2 -9^2 + 5^2 - 1^2.\)



\(\displaystyle S=101^2 - 97^2 + 93^2 - 89^2 + \cdots+13^2 -9^2 + 5^2 - 1^2\)



\(\displaystyle S=(101 - 97)(101 + 97) + (93 - 89)(93 + 89) + \cdots+(13 -9)(13 +9) + (5 - 1)(5 + 1)\)



\(\displaystyle S=792+728 + \cdots+24+88.\)





\(\displaystyle S=792+728 + \cdots+24+88\)

\(\displaystyle +\)

\(\displaystyle S=24+88+\cdots+792+728.\)





\(\displaystyle 2S=816+816+\cdots+816+816.\)



\(\displaystyle 2S=816\cdot13\)



\(\displaystyle S=\frac{816\cdot13}{2}.\)



\(\displaystyle S=5304\)



\(\displaystyle 101^2 - 97^2 + 93^2 - 89^2 + \cdots + 5^2 - 1^2 = \boxed{5304}\)
When pka says "well done" - not once but twice - that is something......
 

homeschool girl

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thanks :)
 

homeschool girl

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i tried doing part b but i kept getting stuck and when i got an answer it was wrong
 
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