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Find 99 - 97 + 95 - 93 + ... + 3 - 1.

 May 3, 2020
 #1
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0

sum_(n=1)^50 (-1)^(n + 1) (101 - 2 n) = 50

 May 3, 2020
 #2
avatar+50 
0

There are 2 ways we can attack this problem...

 

Method 1:

 

Notice that the subtraction result for every 2 pairs has the answer of 2.

 

Since there are 50 pairs that have the result of 2 this is 50(2) or 100.

 

Method 2:

 

We can use Gaussean Addition...

 

The normally the formula we would use is n(n+1)/2 but we are adding all the added pairs, all the subtracted pairs and then subtracting them (if that made any sense...). Since 99+3 = 102 we multiply that by 50/2 or 25. 102 x 25 = 2550.

 

Each pair from the subtracting side sums up to 98. We multiply 98 x 25 to get 2450. 2550 - 2450 = 100.

 

100 is my final answer...

 May 4, 2020
 #3
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Add them all up:

 

(99, -97, 95, -93, 91, -89, 87, -85, 83, -81, 79, -77, 75, -73, 71, -69, 67, -65, 63, -61, 59, -57, 55, -53, 51, -49, 47, -45, 43, -41, 39, -37, 35, -33, 31, -29, 27, -25, 23, -21, 19, -17, 15, -13, 11, -9, 7, -5, 3, -1) = 50

Guest May 4, 2020
 #4
avatar+24952 
+1

Find

\(99 - 97 + 95 - 93 + \ldots+ 3 - 1\).

 

\(\begin{array}{|l|rc|} \hline & \text{pair} \\ \hline 1 & 3-1 & = 2 \\ 2 & 7-5 & = 2 \\ 3 & 11-9 & = 2 \\ 4 & 15-13 & = 2 \\ 5 & 19-17 & = 2 \\ 6 & 23-21 & = 2 \\ 7 & 27-25 & = 2 \\ 8 & 31-29 & = 2 \\ 9 & 35-33 & = 2 \\ 10 & 39-37 & = 2 \\ 11 & 43-41 & = 2 \\ 12 & 47-45 & = 2 \\ 13 & 51-49 & = 2 \\ 14 & 55-53 & = 2 \\ 15 & 59-57 & = 2 \\ 16 & 63-61 & = 2 \\ 17 & 67-65 & = 2 \\ 18 & 71-69 & = 2 \\ 19 & 75-73 & = 2 \\ 20 & 79-77 & = 2 \\ 21 & 83-81 & = 2 \\ 22 & 87-85 & = 2 \\ 23 & 91-89 & = 2 \\ 24 & 95-93 & = 2 \\ 25 & 99-97 & = 2 \\ \hline \end{array}\)

 

\(99 - 97 + 95 - 93 + \ldots+ 3 - 1 = \mathbf{25 \times 2 = 50}\)

 

laugh

 May 4, 2020

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