SOLUTION: Hi im having so much trouble proving this formula with mathematical induction. Please help! {{{ 1/(1*3)+1/(3*5)}}}+...+{{{+1/((2n-1)(2n+1))=n/(2n+1) }}} Basis step: n=1 was

Algebra ->  Real-numbers -> SOLUTION: Hi im having so much trouble proving this formula with mathematical induction. Please help! {{{ 1/(1*3)+1/(3*5)}}}+...+{{{+1/((2n-1)(2n+1))=n/(2n+1) }}} Basis step: n=1 was       Log On


   

Question 1021838: Hi im having so much trouble proving this formula with mathematical induction.
Please help!
+1%2F%281%2A3%29%2B1%2F%283%2A5%29+...+%2B1%2F%28%282n-1%29%282n%2B1%29%29=n%2F%282n%2B1%29+
Basis step: n=1 was true
Assumption step: n=k
%2B1%2F%28%282k-1%29%282k%2B1%29%29=k%2F%282k%2B1%29+
Induction step: Prove true for n=(k+1)
+1%2F%281%2A3%29%2B1%2F%283%2A5%29+...+%2B1%2F%28%282k-1%29%282k%2B1%29%29+
+1%2F%281%2A3%29%2B1%2F%283%2A5%29+...+k%2F%282k%2B1%29++%2B1%2F%28%282k%2B1%29%282k%2B3%29%29=%28k%2B1%29%2F%282k%2B3%29+
From here I have no idea what to do, to the LHS to make it equal to RHS

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
The assumption of induction is:
1%2F%281%2A3%29%2B+1%2F%283%2A5%29+...+1%2F%28%282k-1%29%282k%2B1%29%29=k%2F%282k%2B1%29+
Now add 1%2F%28%282k%2B1%29%282k%2B3%29%29 to BOTH sides.
==> 1%2F%281%2A3%29%2B+1%2F%283%2A5%29+...+
(Note: 1) The last equation is TRUE by the induction hypothesis. 2) We added 1/(2k+1)(2k+3) because it is supposed to be the next term in line after 1/(2k-1)(2k+1).)
The RHS of the last equation is equal to
%28k%282k%2B3%29%29%2F%28%282k%2B1%29%282k%2B3%29%29+%2B+1%2F%28%282k%2B1%29%282k%2B3%29%29
=%282k%5E2%2B3k%2B1%29%2F%28%282k%2B1%29%282k%2B3%29%29
=%28%282k%2B1%29%28k%2B1%29%29%2F%28%282k%2B1%29%282k%2B3%29%29 = %28k%2B1%29%2F%282k%2B3%29

I leave the rest of the inductive argument to you...