What is the nth Term?

Date: 08/02/97 at 23:22:42
From: Chris
Subject: nth term

We are doing math over the summer. In the book Pre-Algebra by Tri-C 
Publications, on page 9 question 7 is: 

The first four terms of a sequence are 16, 8, 4, 2. 

Question A: Find the next two terms. 

   My answer is: 1, and .5. 

Question B: Find the rule for the nth term. 

   What is the nth term? 

Date: 08/08/97 at 20:33:07
From: Doctor Marko
Subject: Re: nth term

Hi Chris,

I believe that you are right about the next two terms in the sequence.  
Now the nth term part of the question means: Find a formula such that 
if I say I want the nth (say 12th or 117th) term of the sequence you 
can give me the formula by which I obtain that term without obtaining 
all the terms ahead of it.

To illustrate, it was easy for you to obtain .5 as the 6th (n = 6) 
term of the sequence because you knew the previous five.  But what if 
I gave you the first three terms and said: "Figure out the sixth term 
without computing the 4th and the 5th!"

Doing so involves a little bit of imagination. For example in your 
question we see that you go from one term to the next by dividing 
by 2, so

   the 1st term (16) must be 32/2
   the 2nd term (8) must be 32/2/2 = 32/4 = 32/(2)^2
   the 3rd term is 4 = 32/2/2/2=32/8 = 32/(2)^3

and so on... 

From this you can see a pattern that says:

   the nth term is equal to 32/(2)^n.

Moreover, you can recall that 32 is a power of 2 as well (2^5), so you 
can write the nth term as: 2^(5-n) - I hope you do not mind this bit 
of exponent review :)

But the question still remains: Does it work? Well, let us try your 
sixth term   .5: 32/(2)^6 = 32/64 = .5.

Voila, and so on...

Hope that this helped,

-Doctor Marko,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/