Question

Show the sum of first p even numbers is equal to (1+1/p) times sum of first p odd numbers.​

Answer 1

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Answer 2

Step-by-step explanation:

Sum of the first p even numbers = 2+4+6+8.....p numbers = 2(1+2+3+4+......p numbers) = 2xp(p+1)/2=p(p+1) ---(1)

Sum of the first odd numbers = 1+3+5+7+...p numbers

= [1+2+3+4+5+6+7.....{p+(p-1)or(2p-1) first natural numbers}] - [2+4+6........(p-1) even numbers]

= [(2p-1)(2p)/2]-[(p-1)(p)(use result (1)] = [2p^2-p]-[p^2-p]=p^2 ------ (2)

Now {1+(1/p)} * sum of first p odd numbers

= {1+(1/p)}*p^2=p^2+p=p(p+1) sum of first p even numbers ...

Hope this will help you. .