Question

Use induction to show that 1+3+5+...+(2N-1)=N^2

Answer 1

Base case \(n=1\).
There is only one term to add so...
We need to show that \(2(\color{red}{1})-1=(\color{red}1)^2\)

Answer 2

hi?
do you know how to do a proof by induction?
summation ones are the easiest

Answer 3

so, it would be 1+3+5+..+(2(1)-1)-1=1^2 @wio

Answer 4

You assume:\[
1+3+5+\ldots+(2N-1)=N^2
\]And then add the next term:\[
1+3+5+\ldots+(2N-1)+(2N+1)=N^2 + (2N+1)
\]And show that this somehow becomes: \[
1+3+5+\ldots+(2(N+1)-1)=(N+1)^2

\]

Answer 5

@sar12389 Yeah, but you don't have the \(1+3+5\) because in the base case, there is only the \(1\) term because \(n\) is too small to reach those terms

Answer 6

so it would just be (2(n+1)-1)=n+1^2 @wio

Answer 7

@wio ?^