# Geometric progression involving logs!

• Apr 24th 2007, 04:02 PM
LoveDeathCab
Geometric progression involving logs!
Hey, can anyone help me with this please?

5. In a geometric progression, the first term is 5 and the second term is 4.8
i) I have solved this. r = 0.96
ii) The sum of the first n terms is greater than 124. Show that

0.96^n < 0.008,

and use logarithms to calculate the smallest possible value of n.

Thank you if you can help me with this. :D
• Apr 24th 2007, 04:22 PM
Jhevon
Quote:

Originally Posted by LoveDeathCab
Hey, can anyone help me with this please?

5. In a geometric progression, the first term is 5 and the second term is 4.8
i) I have solved this. r = 0.96

a geometric progression is one in which the terms are given by the formula:
a_n = ar^(n - 1) for n = 1,2,3,4,5...
where a_n is the nth term, a is the first term and r is the common ration

we are told that the first term of a particular geometric sequence is 5 and the second term is 4.8

now r is given by r = (a_{n+1})/(a_n)
in particular, r = (a_2)/(a_1) = 4.8/5 = 0.96

you are correct:D

Quote:

ii) The sum of the first n terms is greater than 124. Show that

0.96^n < 0.008,

and use logarithms to calculate the smallest possible value of n.
The sum of the first n terms of a geometric sequence/progression is given by:
S_n = a(1 - r^n)/(1 - r)
we are told that S_n > 124
=> a(1 - r^n)/(1 - r) > 124
=> 5(1 - (0.96)^n)/(1 - 0.96) > 124
=>5(1 - (0.96)^n)/(0.04) > 124
=> 5(1 - (0.96)^n) > 4.96 ..............i multiplied through by 0.04
=> 1 - (0.96)^n > 0.992 ..................i divided through by 5
=> -(0.96)^n > -0.008 ....................i subtracted 1 from both sides
=> (0.96)^n < 0.008 ...............................i multiplied through by -1, so i flipped the inequality sign.

Now we will use logs to find the smallest possible n.

0.96^n < 0.008
take log to the base 10 of both sides
=> log(0.96^n) < log(0.008)
=> nlog(0.96) < log(0.008)
=> n > log(0.008)/log(0.96) ...........note that log(0.96) is negative, so when i divided by it, i flipped the inequality sign
=> n > 118.27

so the smallest possible value for n is 119