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Sequences


Date: 7/29/96 at 10:36:50
From: Suryadie Gemilang
Subject: Sequences

Three numbers are a geometric sequence. The sum of the three numbers 
is 147 and the numbers when multiplied together yield the result 
21952. Find the value of each number.

Find the formula for the n-th term of these sequences:
   60, 30, 20, 15, 12, 10,...

Thank you very much
Sg


Date: 7/29/96 at 13:2:40
From: Doctor Paul
Subject: Re: Sequences, etc

What do you know about going from one term to the next in a 
geometric sequence?  You multiply the first term 'a' by some number 
'r' right?  And then you multiply that number by r again to get the 
next term in the sequence.

Let's set up two equations:

       a + a*r + (a*r)*r = 147
 (a) * (a*r) * ((a*r)*r) = 21952

Let's simplify:

         a + a*r + a*r^2 = 147
                 a^3*r^3 = 21952

Let's take the second equation and solve for r

                     r^3 = 21952 / a^3

Take the cube root of both sides and get:

                       r = 28 / a

now let's substitute that into the first equation:

a + a*(28/a) + (a*(28/a)^2) = 147

simplify:

         a + 28 + (784 / a) = 147

multiply through by a:

           a^2 + 28*a + 784 = 147*a

bring the 147*a over to yield:

          a^2 - 119*a + 784 = 0

use quadratic formula:

                          a = 7,112


Now we take these values for a and plug them into one of the two 
equations above to solve for r.   Which equation you choose doesn't 
matter.  I choose the second equation:

7^3 * r^3 = 21952
      r^3 = 64
so      r = 4

now  if a = 112, let's find r

112^3 * r^3 = 21952
        r^3 = 1/65
          r = 1/4

So we have two answers:  a = 112, r = .25  *and* a = 7, r = 4

Now let's go back to how we get the numbers for a geometric sequence.  
We start with a number a and multiply each term by r to get the next 
term.  Let's start with a = 112.

That would yield the set: {112, 28, 7}

if a = 7, you get: {7, 28, 112}

Interesting, huh?  So your three numbers are 7, 28, and 112.


Next problem?

60, 30, 20, 15, 12, 10, x

To move from the first term to the second term, multiply by 1/2.

To go from 30 to 20, multiply by 2/3  To get the next term, multiply 
by 3/4 See what's happening?  Each time you add one to the numerator 
and the denominator and then multiply.  Let's continue:
15 * 4/5 = 12
12 * 5/6 = 10
10 * 6/7 = 60/7 = 8.5714

-Doctor Paul,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Sequences, Series

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