Given the following finite sum
1/(1*2)+1/(2*3)+1/(3*4)+1/(4*5)+....+1/(n(n+1))
a) Find the first 5 partial sums
b) Make a conjecture for a formula for the sum of the first n terms
c) Use mathematical induction to prove your formula
Given the following finite sum
1/(1*2)+1/(2*3)+1/(3*4)+1/(4*5)+....+1/(n(n+1))
a) Find the first 5 partial sums
b) Make a conjecture for a formula for the sum of the first n terms
c) Use mathematical induction to prove your formula
a) You can do.
b) $\displaystyle \sum_{j=1}^n \frac{1}{j(j+1)} = 1 - \frac{1}{n+1}$
c) Do you know what induction is?
Prove base case. I.e. show for n=1,
Assume true for n.
Prove for n+1. Hint, $\displaystyle \sum_{j=1}^{n+1} \frac{1}{j(j+1)} = \sum_{j=1}^n \frac{1}{j(j+1)} + \frac{1}{(n+1)(n+2)}$
No you assume its true for n=k then prove it for n=k+1.
http://en.wikipedia.org/wiki/Mathematical_induction