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

Question

in 1992, the number of units sold 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 trend in sales continues.

 

(b) How many units will be sold during 2002?

 

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

 

Between 1990 and 1992, the total number of units sold is 760. (d) What is the total 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. 

Answer 1

(a)  Common ratio is calculated by taking the current number and dividing it by the previous number.  In geometric sequences, the ratio is always the same and is denoted as r.

 

240/160 = 3/2 or 1.5      360/240 = 3/2 or 1.5 

 

therefore r = 1.5

 

(b)  To answer this question, we remember that the formula for any term in a geometric sequence is a_n=a₁·r^n-1 where n is the term number and a₁ is the first term of the sequence.  2002 would be the 13th term in the sequence, so

 

a₁₃ = 160 · 1.5^13-1 = 29759.41 or 29759 units

 

(c)  This question is accomplished by guessing and checking values.  Due to the answer to part b, we know that 5000 happens earlier.  In 1998, the total is 4100.625 while in 1999, the total is 6150.94, so 1999 is the first year the number passes 5000 units.

 

(d)  One way to calculate this is to calculate each term individually and then add up.  There is also a formula for a finite partial sum of a geometric sequence: (a₁-a_nr)/(1-r)

 

We already know a₁ and a₁₃ and r, so its a simple matter of plug and chug.

 

(160 - 29759.41(1.5))/(1-1.5) = 61958.24 or 61958 units

 

(e)  If the population was 80,000, then eventually everyone in the population would have a cell phone and there'd be no one else to sell to.  The geometric growth would have to stall out because of that.