Expert: Josh - 6/8/2009

Question
Please help.
The Question:
The sum of the first three terms of a geometric sequence of positive integers is equal to seven times the first term and the sum of the first four terms is forty five. What is the first term of the sequence.
This is what I have;
a+b+c=7a where a=1st term, b=2nd term and c=3rd term
a+b+c+d=45
Am i going in the correct direction? My step will be to combine both equations yielding 2a+2b+2c+d=7a+45. I think??

Answer
Jackie,

Re: a+b+c=7a ...[1] and a+b+c+d=45 ...[2], what you said is true. To go further, we need to recognize that "a","b" and "c" are not just any unknown quantities; they in fact represent the first three terms of a geometric sequence.

If R represents the ratio between successive terms. Then, we have b/a=R, c/b=R, d/c=R etc. This gives: b=aR, c=aR^2, d=aR^3.

Thus, there are actually only two unknowns "a" and "R" which govern the system of equations.

Using this relationship, we write the equation [1] as a+aR+aR^2=7a and factorize it as (1+R+R^2)=7 ...[3]. Similarly, [2] is turned into
a(1+R+R^2+R^3)=45 ....[4]

Solving the quadratic equation of [3], we get the value of R. This can be done almost by inspection, since we were told it must be a positive integer. Substituting into [4] we get the value of the first term "a".