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(a) Compute the sum

 

 \(101^2 - 97^2 + 93^2 - 89^2 + \cdots + 5^2 - 1^2.\)

 

(b) Compute the sum

 

  \((a +(2n+1)d)^2- (a + (2n)d)^2 +(a + (2n-1)d)^2 - (a+(2n-2)d)^2 + \cdots + (a+d)^2 - a^2.\)

 

 

Repost of this one:

https://web2.0calc.com/questions/need-help_24323

 Apr 21, 2020
edited by Melody  Apr 21, 2020
 #1
avatar+57 
+1

Are you posting every question of your homework on here? Your not going to learn anything by making us do it for you. Maybe you should have a go at your homework questions yourself first before asking us to do it for you. It's ok to ask for help if you truly get stuck, but you shouldn't be asking us to do your whole homework for you. Who knows; you might get all the answers to your questions yourself if you have a go at them. I'm sure whoever set you your homework won't have set you anything you couldn't already do. Have a go at them. Good luck smiley

 Apr 21, 2020
 #2
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No. This is the first question I have posted to this site.

I am asking for help with a proper explanation so that I may understand the problems and their solutions better.

I am under the "guest" profile because I haven't made an account and I don't feel the need to.

If you don't want to answer, that is fine, but I would appreciate if you could at least point me in the right direction.

 

Thank you.

Guest Apr 21, 2020
 #3
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*by first, I mean the first question I have posted to this site, though I have asked it once before on a different post.

Guest Apr 21, 2020
 #4
avatar+109492 
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Please include the link to your first post.

 Apr 21, 2020
 #5
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https://web2.0calc.com/questions/need-help_24323

Guest Apr 21, 2020
 #6
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+1

a - The answer of 5,304 in NOT wrong! It just wasn't explained to you how they arrived at the answer. What is the difference between 1st term and 2nd term? 101 - 97 =4. Each term after the first is 4 LESS than the previous term.So, you get: 101 - 4 =97 - 4 =93 - 4 =89 - 4=85.........etc. all the way down to 5 - 4 =1. And then each term is "squared":


101^2 =10,201 - 97^2 =9,409 + 93^2 =8,649......etc.
This is the "closed form formula" that generates each term of your sequence:  (-1)^(n+1)*(105 - 4*n)^2, where n=1 to 26.


And this is what you get, when you expand your sequence: [10201, -9409, 8649, -7921, 7225, -6561, 5929, -5329, 4761, -4225, 3721, -3249, 2809, -2401, 2025, -1681, 1369, -1089, 841, -625, 441, -289, 169, -81, 25, -1]
Sum them up on your calculator term by term and you should get =5,304.
I hope you understand it now. Good luck to you.

Guest Apr 21, 2020
edited by Guest  Apr 21, 2020
 #7
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Oof. I did not understand the problem at all then. Thank you for clearing that up! Sorry about that.

Guest Apr 21, 2020
 #10
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Take the difference in squares and then sum them up as an arithmetic sequence as follows:


All you need are the first term [101^2 - 97^2]=792 and the last term [5^2 - 1^2] =24 and the number of terms = 26 and the common difference = 4

                    
Sum =[F + L] / D x N, where F=First term, L=Last term, D=Common difference, N=number of terms.


Sum =[792 + 24] / 4 x 26
        =[816 / 4] x 26
        = 204 x 26
        = 5,304

Guest Apr 21, 2020
 #8
avatar+29978 
+2

You can look at this question as follows:

 

 Apr 21, 2020
 #9
avatar+29978 
+2

Here's the answer to part b:

 

 

You should check this carefully, as I did it rather hurriedly!

 Apr 21, 2020

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