Question

Prove that 2+4+6+.....+2ⁿ=n(n+1)

Answer 1

We will employ the sum of arithmetic progression to prove our series.

`s=(n)/(2)(2a+(n-1)d`

where:

s = sum of the series = 2 + 4 +6 + 8...+2n

n = arbitrary term

a = first term = 2

d = common difference = 6 - 4 = 4 - 2 = 2

`\begin{gathered} s=(n)/(2)(2(2)+(n-1)2) \\ We\text{ expand the expression to get:} \\ s=(n)/(2)(4+2n-2) \\ s=(n)/(2)(2n+2) \\ s=n((2n)/(2)+(2)/(2)) \\ s=n(n+1) \end{gathered}`