Question

Use Mathematical Induction to show that the statement
2 + 6 + 10 + . . . + (4n – 2) = 2n^2
is true

Answer 1

n an sum n
1 2 2
2 6 8
3 10 18
4 14 32

n (4n-2) 2n^2

(n+1) (4(n+1)-2) [2n^2+ (4(n+1)-2)] or hopefully 2(n+1)^2

[2n^2+ (4(n+1)-2)] = 2n^2+4n +2
and
2(n+1)^2 = 2(n^2+2n+1) = 2n^2+4n+2 done

2n^2+ (4(n+1)-2)

Answer 2

Using the 3 step process:
1. test for n=1
LS = 2
RS = 2(1^2) = 2

2. assume it true for n=k
that is ....
2+4+6 + ... + (4k-2) = 2k^2

3 . prove it is then true for =k+1
or
2+4+6+ ... + (4k-2) + 4(k+1)-2 = 2(k+1)2
LS = [2+4+6+...+ 4k-2 ] + 4(k+1)-2
= 2k^2 + 4k + 4 - 2
= 2(k^2 + 2k + 1)
= 2(k+1)^2
= RS

QED