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Compute the sum 1012972+932892++5212 by pairing the terms and applying the difference of squares.

 Jul 9, 2020
edited by mathmathj28  Jul 9, 2020
edited by mathmathj28  Jul 9, 2020
edited by mathmathj28  Jul 9, 2020
 #1
avatar+23254 
+3

1012 - 972  =  (101 - 97)(101 + 97)  =  4·198

  932 - 892  =  (93 - 89)(93 + 89)      =  4·182

  852 - 812  =  (85 - 81()(85 + 81)     =  4·166

   ...

      52 - 12  =  (5 -1)(5 + 1)               =  4·6 

 

Sum  =  4·198  +  4·182  +  4·166  +  ...  +  4·6  =  4·[ 198 + 182 + 166 + ... + 6 ]

 

198 + 182 + 166 + ... + 6  is an arithmetic series whose first term = 198, last term = 6, common difference =  -16

 

The formula for the sum is:  Sum  =  n · (t1 + tn) / 2

 

We'll need to find the number of terms; we can do this by using this formula:  tn  =  t1 + (n - 1)·d

   --->   6  =  198 + (n - 1)·-16   --->   -192  =  (n - 1)·-16   --->   12  =  n - 1   --->   n  =  13

 

Sum  =  n · (t1 + tn) / 2   --->   Sum  =  13 · (198 + 6) / 2    --->   1326

 

So, 4·[ 198 + 182 + 166 + ... + 6 ]  =  4·1326  =  5304

 Jul 9, 2020
 #2
avatar+306 
+2

Thank you so much for the explanation!!!

mathmathj28  Jul 9, 2020

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