Answer to Question #147160 in Algebra for Ruchika

Question

Prove that the sum of the first n even numbers is an even number. Is the kind of logic used in proving this is inductive, deductive or both? Justify your answer.

Answer 1

Proof

Let n= 2k for some integer k

Then, in a series of even integers we obtain

$2+4+6+...+2k=k(k+1)$

$P(k)=1(1+1)=2,$ where k=1

$P(k+1)=2+...+2(k+1)$

$=k(k+1)+2(k+1)$

$=k^2+k+2k+2$

$=k^2+3k+2$

$=k(k+1)[(k+1)+1]$ Q.E.D

This shows the prove that the sun of n even numbers is even.

This is inductive proof.

The inductive step involves assuming the statement is true for one number, then proving it for the next number. ... If somehow integers came in two chunks with 0 in the first chunk, then mathematical induction would only work to prove that the statement is true in the first chunk.