Prove by math induction that 1+3+5+7+.......+(2n-1)=n²?

1 Answer
Lucy
Apr 3, 2018

Step 1: Prove true for n=1
LHS= 2-1=1
RHS=1^2= 1= LHS
Therefore, true for n=1

Step 2: Assume true for n=k, where k is an integer and greater than or equal to 1

1+3+5+7+....+(2k-1)=k^2 ------- (1)

Step3: When n=k+1,
RTP: 1+3+5+7+...+(2k-1)+(2k+1)=(k+1)^2

LHS:
1+3+5+7+...+(2k-1)+(2k+1)
=k^2+(2k+1) ---(from 1 by assumption)
=(k+1)^2
=RHS
Therefore, true for n=k+1

Step 4: By proof of mathematical induction, this statement is true for all integers greater than or equal to 1

(here, it actually depends on what your school tells you because different schools have different ways of setting out the final step but you get the gist of it)

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