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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