Thread: Proof by Mathematical Induction

Prove that for all integers n >= 1,

1/3 = (1 + 3)/(5 + 7) = (1 + 3 + 5)/(7 + 9 + 11) = ...
= (1 + 3 + ... + (2n - 1))/((2n + 1) + ... + (4n - 1))

This is what I have so far:
Show that P(1) is true. Left-hand side = 1/3. Right-hand side = 1/3. Since the left-hand side is equal to the right-hand side, P(1) is true. Show that for any integer k >= 1, if P(k) is true, then P(k + 1) is also true.
Suppose that k is any integer with k >= 1.

I am confused about how to complete the inductive step. Can anyone help me with this?

Originally Posted by lovesmath

Prove that for all integers n >= 1,
1/3 = (1 + 3)/(5 + 7) = (1 + 3 + 5)/(7 + 9 + 11) = ...
= (1 + 3 + ... + (2n - 1))/((2n + 1) + ... + (4n - 1))

I would do it in two parts.
The numerator .

The denominator

The separate inductions should be easier.