H2 mathematical induction

wishboy SG

A sequence is defined by u(1) = 1 and u(n+1) = u(n) / [u(n) + 2] for all positive integers of n.

Write down the first five terms of the sequence {1/u(n)}.

my answer: {1, 3, 5/3, 11/5, 21/11}

Hence make a conjecture for u(n) and prove your conjecture using induction.

this is the part where i'm stuck at. how do i go about making the conjecture?

wee_ws SG

Hi,

Please check your work again.

I got the following:

u_2 = 1/3, so 1/u_2 = 3 = 2^2 - 1;

u_3 = 1/7, so 1/u_3 = 7 = 2^3 - 1;

u_4 = 1/15, so 1/u_4 = 15 = 2^4 - 1;

u_5 = 1/31, so 1/u_5 = 31 = 2^5 - 1;

u_6 = 1/63, so 1/u_6 = 63 = 2^6 - 1.

Hence 1/u_n = 2^? - 1.

The key to making a conjecture is to look at patterns. If you refer to the answers you have obtained, there does not seem to have a pattern, so you may suspect that something is not right and you ought to look back at your working.

Thanks!

Cheers,
Wen Shih

wee_ws SG

Hi,

The type of conjecture we saw is one of the common types, where it can be expressed as a power of some number, where the power is related to the subscript of u.

Suppose u_1 = 1/2, u_2 = 3/5, u_3 = 2/3, u_4 = 5/7, u_5 = 3/4. This type of conjecture involves a fraction where the numerator and the denominator are related, and they can be expressed in terms of the subscript. If we look closely at the sequence again:

u_1 = 2/4, u_2 = 3/5, u_3 = 4/6, u_4 = 5/7, u_5 = 6/8.

We can now see a pattern more clearly and conclude that u_n = (n + 1) / (n + 3).

I will describe another type in my next posting. Thanks!

Cheers,
Wen Shih

wishboy SG

oh so that was where my mistake was...

thanks mr wee!

wee_ws SG

Hi,

Suppose we have a sequence that looks like this:

u_1 = 5/3, u_2 = 1/3, u_3 = 1/7, u_4 = 5/63, u_5 = 5/99.

We could guess that the numerator is always 5 and the denominator is a product of two factors. The two factors may also be related.

Rewriting the sequence, we obtain

u_1 = 5 / (1)(3),

u_2 = 5 / (3)(5),

u_3 = 5 / (5)(7),

u_4 = 5 / (7)(9),

u_5 = 5 / (9)(11).

It is now possible for us to state the formula

u_n = 5 / (2n - 1)(2n + 1).

Thanks!

P.S. It is useful to remember some results when making your conjecture:

1. 2n is always even.

2. (2n - 1) or (2n + 1) is always odd.

3. (-1)^n = -1 for odd n, while (-1)^n = 1 for even n.

4. n! = n(n - 1)(n - 2)...(3)(2)(1) or n! = n(n - 1)! equivalently.

Cheers,
Wen Shih

wee_ws SG

Hi,

Suppose we have this sequence:

u_1 = (1)(1^2), u_2 = (2)(4^2), u_3 = (3)(7^2), u_4 = (4)(10^2).

Now look at the second terms without powers, i.e. 1, 4, 7, 10. We can see an arithmetic sequence with first term 1 and common difference 3.

Thus, we are able to make the conjecture u_r = (r) (3r - 2)^2.

Thanks!

P.S. It is helpful to remember AP/GP concepts when making conjectures.

Cheers,
Wen Shih