# Arithmetic Sequences

#### homeschool girl

##### Junior Member
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!

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#### Dr.Peterson

##### Elite Member
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.)

5304 for part a?

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#### homeschool girl

##### Junior Member
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

##### Elite Member
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

##### Super Moderator
Staff member
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......

thanks

#### homeschool girl

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