Use induction to prove the following equation: $2 + 6 + 10 + \cdots + (4n − 2) = 2n^2$ where $n \ge 1$

Question

Use induction to prove the following equation: $2 + 6 + 10 + \cdots + (4n − 2) = 2n^2$

where $n \ge 1$

Answer 1

Let $$p(n):2+6+10+\cdots+(4n-2) = 2n^2\;,$$ Put $n=1\;,$ We get $2=2$

so $p(n)$ is true for $n=1$

Now Put $n=k\;,$ We get $$p(k):2+6+10+\cdots+(4k-2) = 2k^2\;$$

Now Put $n=k+1\;,$ We get $$P(k+1): 2+6+10+\cdots+(4k+2)$$

Now Using $p(k)$ we will prove for $p(k+1)$

So we can write $$p(k+1): 2+6+10+\cdots+(4k-2)+(4k+2)= 2k^2+4k+2 = 2(k+1)^2$$

So it is True for $p(k+1)$