Question

1.4 ⋅6 + 5 ⋅7 + 6 ⋅ 8 + ... + 4n( 4n + 2) =4(4n+1)(8n+7)/6

Answer 1

Mathematical induction

Answer 2

Let n = k, and assume that the sum is correct:

S(k) = 4*6 + 5*7 + ... + 4k(4k + 2) = 4(4k+1)(8k+7) / 6

Now, show that if it is true for k, then it is also true for k+1:
S(k+1) = 4*6 + 5*7 + ... + 4k(4k+2) + 4(k+1)(4(k+1)+2) = 4(4(k+1)+1)(8(k+1)+7) / 6

We can see that S(k+1) is just S(k) with an extra term:
S(k+1) = S(k) + 4(k+1)(4(k+1)+2)

Since we're assuming the equation 4(4n+1)(8n+7)/6 is true for S(k), and we're showing that it then is also true for k+1, we can replace S(k) and S(k+1):
4(4(k+1)+1)(8(k+1)+7) / 6 = [4(4k+1)(8k+7) / 6] + 4(k+1)(4(k+1)+2)

Show, with algebra, that that must be true. Then, prove that the original sum is true for n=1

Answer 3

CanI make that I understaand this by showing you another equation by me myself doing and you can help me along the way?

Answer 4

@DDCamp