# Math Help - sequence & Series

1. ## sequence & Series

hey guys, i hav the answers to this question which ive written in red, but i need to know the working out....

Portable telephones are first sold in the country Cellmania in 1990. During 1990, the number of units sold is 160. in 1991, the number of units sold is 240 and in 1992 it is 360.

In 1993 it was noticed that the annual sales formed a geometric sequence with first term 160, the 2nd and 3rd terms being 240 and 360 respectively.

a) what is the common ratio of this sequence?

Assume that this sales trend continues.

b) how many units will be sold during 2002?

c) in what year does the number of units sold between 1990 and 2002?

During this period, the total population of Cellmania remains approximately 80,000

e) use this information to suggest a reason why the geometric growth in sales would not continue

a) 1.5
b) 20,759
c) 1999
d) 61,958

2. a) A geometric series is a series with a constant (common) ratio between successive terms. Therefore, if 160 is the first term and the second term is 240, then the common ratio is $\frac{240}{160} = \frac{3}{2} = 1.5$.

b) The $n$th term of a geometric sequence with initial value $a$ and common ratio $r$ is given by $a_n = ar^{n - 1}$. If we consider 1990 as the initial year, then $a = 160$ and $r = 1.5$. Therefore, the number of units sold during 2002 will be given by the 13th term. Why do we use 13 instead of 12 (since 2002 is 12 years after 1990)? The question asks how many units will be sold during 2002, which includes the year 2002. Another way to word the question would be how many units will be sold by 2003. Returning to our formula for the nth term of a geometric sequence, the 13th term of our sequence is given by $a_13 = 160 * 1.5^{13 - 1} = 20,759.4141$, which rounds to 20,759.

c) I believe you omitted the last few words of the question.

3. Yes thank you so much for a & b, i completely understand, and yeah i mixed up the question after that...

c) in what year does the number of units sold first exceed 5000?

d) what is the total number of units sold between 1990 and 2002?

e) use this info to suggest a reason why the geometric growth in sales would not continue...

Thanks alot for a & b though

4. c) We already know the $n$th term of geometric sequence with initial value $a$and common ratio $r$ is given by $a_n = ar^{n - 1}$. In this case, a is still 160 and r is still 1.5. The question asks when does the number of units sold ( $a_n$) exceed 5000, or $a_n > 5000$. Substitute $a_n = 160 \times 1.5^{n - 1}$ and solve for n. (Remember the n is a number of years after 1990.)

d) The sum of the first $n$ terms of a geometric sequence with initial value $a$ and common ratio $r$ is given by $S_n = \frac{a(1 - r)^n}{1 - r}$. In this case, a is still 160 and r is still 1.5. Therefore, the sum of the first 13 terms of our sequence is given by $S_13 = \frac{160(1 - 1.5^{13})}{1 - 1.5} = 61,958.2422$, which rounds to 61,958.