Answer to Question #87702 in Calculus for Adri

Question

Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false:
2+6+10+...+4n-2=2n^2

Answer 1

If n=1 , then statement is true. Denote a(n)=2+4+10+...+(4n-2), b(n)=2n^2. 

Assume that the statement is true for n : a(n)=b(n). We show that it follows from here that

a(n+1)=b(n+1), a(n)=b(n)→a(n+1)=a(n)+(4n+4-2)=a(n)+4n+2=b(n)+4n+2=

=2n^2+4n+2=2(n+1)^2=b(n+1).The statement is true. 

Pay attetion in the left part there are no brackets (4n-2). Without brackets the expression in the

left part is written incorrectly.