SOLUTION: A geometric progression and an arithmetic progression have the same first term. The second and third terms of the geometric progression, which are distinct, are equal to the third

Question 1083343: A geometric progression and an arithmetic progression have the same first term. The second and third terms of the geometric progression, which are distinct, are equal to the third and fourth terms of the arithmetic progression, respectively.
(1) Find the common ratio of the geometric progression.
(2) Show that the fifth term of the arithmetic progression is zero.
Answer by htmentor(1343) (Show Source): You can put this solution on YOUR website!
The nth term of the arithmetic progression is a_n = a + (n-1)d, where a is the 1st term and d is the common difference
The nth term of the geometric progression is a_n = ar^(n-1), where a is the 1st term and r is the common ratio.
Second term of the G.P. = third term of the A.P. -> ar = a + 2d [1]
Third term of the G.P. = fourth term of the A.P. -> ar^2 = a + 3d [2]
Eliminate d by multiplying [1] by 3/2 and subtracting [1] from [2]:
ar^2 - (3/2)ar = a - (3/2)a
Dividing through by a and simplifying gives:
r^2 - (3/2)r + 1/2 = 0 -> 2r^2 - 3r + 1 = 0
This gives two solutions, r = 1 and r = 1/2
r = 1 makes all terms equal to the first term, so we choose the 2nd solution
Ans: r = 1/2
Using [1] and the value for r, we have:
(1/2)a = a + 2d -> a = -4d
The 5th term of the A.P. is a_5 = a + 4d -> a_5 = -4d + 4d = 0

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