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Alternating Sum of Consecutive Odd and Even Products (Posted on 2024-07-07)

Using only paper and pencil, determine the value of

(101 × 99) - (102 × 98) + (103 × 97) − (104 × 96) + ... + (149 × 51) − (150 × 50).

No Solution Yet Submitted by Danish Ahmed Khan

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Possible Solution

| Comment 1 of 4

Note:

(101*99)=(100-1)(100+1) = 100^2-1

(102*98)=(100+2)(100-2) = 100^2-4 etc.

Generally, the 100^2 terms cancel, while the even square terms become positive.

So, for n terms we have -1+4-9+16-25...

Working:

Sum of first n odd squares: 1/3n(4n^2-1)

Sum of first n even squares: 2/3n(2n^2+3n+1)

Sum of (even-odd) squares 2/3n(2n^2+3n+1)-1/3n(4n^2-1) = 2n^2+n

Here, n=25:

2(625)+25=1275

Posted by broll on 2024-07-08 00:25:10

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