SOLUTION: The sum of the first three terms of a geometric progression (GP) is 38; and the fourth term exceeds the first by 19. find the value of the first term and of the common ratio.
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-> SOLUTION: The sum of the first three terms of a geometric progression (GP) is 38; and the fourth term exceeds the first by 19. find the value of the first term and of the common ratio.
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Question 758723: The sum of the first three terms of a geometric progression (GP) is 38; and the fourth term exceeds the first by 19. find the value of the first term and of the common ratio.
You can put this solution on YOUR website! The nth term of a GP is written a_n = a*r^(n-1), where a=the 1st term, r=common ratio
The sum of the 1st 3 terms = 38 = a + ar + ar^2 = a(1+r+r^2)
The 4th term exceeds the 1st term by 19:
ar^3 = a + 19 -> a(r^3-1) = 19
r^3-1 can be factored as (r-1)(r^2+r+1)
So we have the following two equations:
a(1+r+r^2) = 38
a(r-1)(1+r+r^2) = 19
Divide the 2nd equation by the 1st:
r-1 = 1/2
r = 3/2
The first term = a = 38/(1+3/2+(3/2)^2)) -> a = 38/(19/4) = 8
