I have this question: Find the value of: (1)/(1^2+1)+(1)/(2^2+2)+(1)/(3^2+3)+(1)/(4^2+4)++...+(1)/(2008^2 + 2008)

Massatfy

Massatfy

Answered question

2022-08-11

Find the value of 1 1 2 + 1 + 1 2 2 + 2 + 1 3 2 + 3 + 1 4 2 + 4 + + + 1 2008 2 + 2008
My attempt:
I tried to think of a better way to handle:
1 n 2 + n
Then I got (doesn't work only on 1 1 2 + 1 ):
n 1 n 3 n
By putting the values in, I got:
1 2 + 2 1 2 3 2 + 3 1 3 3 3 + 4 1 4 3 4 + + 2008 1 2008 3 2008
It's still doesn't make sense. Is there another way of solving this question? Can I have a hint or a guide?

Answer & Explanation

alienceenvedsf0

alienceenvedsf0

Beginner2022-08-12Added 15 answers

Write 1 n 2 + n = 1 n ( n + 1 ) = 1 n 1 n + 1 . See if now you can identify a telescoping sequence as follows :
n = 1 2008 1 n 2 + n = n = 1 2008 ( 1 n 1 n + 1 ) = 1 1 2009
Yair Valentine

Yair Valentine

Beginner2022-08-13Added 6 answers

Hint: 1 x ( x + 1 ) = 1 x 1 x + 1 . Now telescope.
You make want to look up partial fraction decomposition, this can be very useful in contest problems, it helps to solve some recurrences via generating functions, and it also helps to find telescoping sums.

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