sum

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

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...

Find

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

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

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