Determine the value of
(101*99) - (102*98) + (103*97) - (104*96) + + (149*51) - (150*50).
(without using calculator)

101 = 100 + 1

102 = 100 + 2

103 = 100 + 3

and so on, and

99 = 100 - 1

98 = 100 - 2

97 = 100 - 3

and so on. Then the -th term of the sum, where , is

We want to compute the sum,

We have

but notice that in the last sum, we're just adding the same number of 1s and -1s together, so its value is 0 and

In case you're not familiar with the formula for the sum of consecutive squares, we can derive it here. Recall that

Notice that

and that

Then

Now consider the cases where is either odd or even.

The original sum is obtained by adding the odd- and even-indexed sums together:

look a the sides they are perpendicular

linear

step-by-step explanation: