Help Mathematicians! Please Solve this problem! Find the product of the first 2007 terms?

Question

The sum of the first n terms in a sequence is always 1/n. Find the product of the first 2007 terms. Show work if you can. Thanks for your help.

Answer 1

Well, let's figure out what the nth term is.

That would be 1/n - 1(n-1) = -1/n(n-1).

At least, that's true for n = 2, 3, ... The first term is 1.

So the product of the first k terms is the product from n=2 to k of -1/n(n-1). And that's just (-1)^(k-1) * 1/k!(k-1)!.

Plug in 2007 for k and you're done.

Answer 2

Simply choose and sub in some numbers to find the terms.

Sub
n = 1, Sum = 1
n = 2, Sum = 1/2
n = 3, Sum =1/3
n = 4, Sum = 1/4

As can be seen,
nth term = Sum of n terms - Sum of (n - 1) terms

Product of first 2 terms
= 1 x (1/2 - 1)

Product of first 3 terms
= 1 x (1/2 -1) x (1/3 - 1/2)

Product of first 4 terms
= 1 x (1/2 - 1) x (1/3 - 1/2) x (1/4 - 1/3)

Product of first n terms
= 1 x (1/2 - 1) x (1/3 - 1/2) x (1/4 - 1/3) x .... x (-1/(n(n - 1)))


2007 1
∏ - ⎯⎯⎯⎯⎯⎯⎯⎯⎯
n=2 n·(n - 1)
= 1.083131729 x10^(1-1.1514x10^4)

Alternatively,
When n is large,
the product becomes approximately 0