Question 1197301: Write the 4 terms of the Geometric Sequence: If the sum of the first and fourth terms is 195, and the sum of the second and the third term is 60
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Let's use the standard notation of a for the first term and r for the common ratio. Then
The sum of the first and fourth terms is 195:
 [1]
The sum of the second and third terms is 60:
 [2]
It should be clear that both a and r are integers, since both sums are integers.
So a quick informal method of solving the problem is to look at equation [1] and find what integer r can be so that (1+r^3) is a factor of 195.
r=1 --> 1+r^3=2 --> not a factor of 195
r=2 --> 1+r^3=9 --> not a factor of 195
r=3 --> 1+r^3=28 --> not a factor of 195
r=4 --> 1+r^3=65 --> YES -- 195 = 3(65)
So it looks as if a=3 and r=4 are the values we are looking for. Let's check that.
The sequence is 3, 12, 48, 192. 3+192 = 195, and 12+48 = 60.
ANSWER: The geometric sequence is 3, 12, 48, and 192.
Note that the sequence could also be 192, 48, 12, 3 -- with first term 192 and common ratio 1/4.
For a formal (and ugly) algebraic solution, using equations [1] and [2] above....
This gives us the two values we found above for the common ratio: r=4 or r=1/4.
And that leads us quickly to the two sequences we found.
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