Use mathematical induction to prove that (in Description below) for all integers n greater than or equal to 1.

Question

Use mathematical induction to prove that (1/(1*3))+(1/(3*5))+(1/(5*7))+...+(1/((2n-1)(2n+1)))=(n/(2n+1)) for all integers n≥1.

Answer 1

n=1:

1/[1*3] = 1/3 = 1/(2*1+1)

n=2:

1/(1*3) + 1/(3*5) = 1/3 + 1/15 = 6/15 = 2/5 = 2/( 2*2+1)

Given that it holds for positive integer k:

1/(1*3) + 1/(3*5) + 1/(5*7) + ... 1/((2n-1)(2n+1)) = n/(2n+1)

1/(1*3) + 1/(3*5) + 1/(5*7) + ... 1/((2n-1)(2n+1)) + 1/((2n+1)(2n+3)) =

n/(2n+1) + 1/((2n+1)(2n+3)) = <--- substitutes induction hypothesis

[n(2n+3) + 1]/((2n+1)(2n+3)) = <--- common denominator (2n+1)(2n+3)

(2n^2 + 3n + 1)/((2n+1)(2n+3)) = <--- combines numerators

(2n + 1)(n + 1)/ ((2n+1)(2n+3)) = <--- factors

(n+1) / (2n+3) = <--- 2n+1 cancels

(n+1) / ( (2n+2)+1 ) = <--- makes it looks like the formula

(n+1) / ( 2(n+1) + 1)