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hicaderasu1c8q 2022-06-04 Answered
1 2 2 2 + 3 2 4 2 + + 1999 2 19992 using closed form formula for sum of squares of first n natural numbers
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Answers (2)

opiftinkistj6mi6
Answered 2022-06-05 Author has 2 answers
If you first add all the squares:
1 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 +
and then subtract the even squares:
1 2 2 2 + 3 2 4 2 + 5 2 6 2
then you are left with only the odd squares:
1 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 +
You need to subtract the even squares one more time to get
1 2 2 2 + 3 2 4 2 + 5 2 6 2 +
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fulgoripb0b
Answered 2022-06-06 Author has 1 answers
We can use the identity
( x + 1 ) 2 x 2 = x + 1 + x
to tackle the sum in 2 simpler ways:
Way 1
1 2 2 2 + 3 2 4 2 + 5 2 . . . 1998 2 + 1999 2 = 1 + ( 2 + 3 ) + ( 4 + 5 ) + + ( 1998 + 1999 ) = 19990000
Way 2
( 1 2 2 2 + 3 2 4 2 + 5 2 6 2 . . . 1998 2 ) + 1999 2 = ( ( 1 + 2 ) + ( 3 + 4 ) + ( 5 + 6 ) + + ( 1997 + 1998 ) ) + 1999 2 = 19990000
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