1.(A)(i)Show that 2+4+6+8+..+2n=n(n+1).

Question

(ii)Find the sum of the first 200 even numbers.
(iii)Find the sum of the first 200 odd numbers.
(B)(i)Use the formula at the beginning of the question to find the sum of the first 2n natural numbers.
(ii)Find a formula, in its simplest form, for 1+3+5+7+9+...+(2n-1)

Answer 1

using induction
if n=1, 2 = 1(2) = 2

assuming true for n=k,
check for n=k+1:
2+...+2k+2k+2
= k(k+1) + 2k+2
= k(k+1) + 2(k+1)
= (k+1)(k+2)
as desired.

1+3+5+...+(2n-1) = n^2