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25 Nov 2021, 03:00
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The sequence of numbers \(a\), \(ar\), \(ar^2\), and \(ar^3\) are in geometric progression. The sum of the first four terms in the series is 5 times the sum of first two terms and \(r\neq{–1}\). How many times larger is the fourth term than the second term?
(A) 1
(B) 2
(C) 4
(D) 5
(E) 6
(A) 1
(B) 2
(C) 4
(D) 5
(E) 6
27 Nov 2021, 11:36
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Given: The sequence of numbers \(a\), \(ar\), \(ar^2\), and \(ar^3\) are in geometric progression. The sum of the first four terms in the series is 5 times the sum of first two terms and \(r\neq{–1}\).
Asked: How many times larger is the fourth term than the second term?
The sum of the first four terms in the series = a + ar + ar^2 + ar^3
The sum of first two terms in the series = a + ar
The sum of the first four terms in the series is 5 times the sum of first two terms and \(r\neq{–1}\).
a + ar + ar^2 + ar^3 = 5 ( a + ar)
ar^2 (1 + r) = 4 a (1 + r)
r^2 = 4
The fourth term in the series = ar^3
The second term in the series = ar
ar^3 / ar = r^2 = 4
IMO C
Asked: How many times larger is the fourth term than the second term?
The sum of the first four terms in the series = a + ar + ar^2 + ar^3
The sum of first two terms in the series = a + ar
The sum of the first four terms in the series is 5 times the sum of first two terms and \(r\neq{–1}\).
a + ar + ar^2 + ar^3 = 5 ( a + ar)
ar^2 (1 + r) = 4 a (1 + r)
r^2 = 4
The fourth term in the series = ar^3
The second term in the series = ar
ar^3 / ar = r^2 = 4
IMO C
