Prove by induction that 1+3+⋯+(2n+1)=(n+1)^2.

Nann

Nann

Answered question

2021-01-28

Prove by induction that 1+3++(2n+1)=(n+1)2.

Answer & Explanation

funblogC

funblogC

Skilled2021-01-29Added 91 answers

Step 1: Show that the statement is true for n=1: 1+3=(1+1)2
4=22
4=4
Step 2:Assume that the formula is true for n=k: 1+3++(2k+1)=(k+1)2
Show that it is truer for n=k+1
1+3++(2k+1)+[2(k+1)+1]=[(k+1)+1]2
1+3++(2k+1)+(2k+3)=(k+2)2
But 1+3++(2k+1)=(k+1)2 so:
(k+1)2+(2k+3)=(k+2)2
Expand both sides: (k2+2k+1)+(2k+3)=k2+4k+4
k2+4k+4=k2+4k+4
Since the equition is true for all k, then the statement 1+3++(2n+1)=(n+1)2
is true for all positive integers n

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