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The sequence of numbers a, ar, ar2, and ar3 are in geometric
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02 May 2019, 02:48
02 May 2019, 02:48
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Question Stats:
60% (02:26) correct
40% (02:25) wrong based on 30 sessions
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The sequence of numbers \(a\), \(a\)r, \(ar^2\), 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 \(r \neq –1\) and \(a \neq 0\). 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
Re: The sequence of numbers a, ar, ar2, and ar3 are in geometric
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16 Apr 2020, 02:34
16 Apr 2020, 02:34
1
Expert Reply
So, we have this sequence: \(a, ar, ar^2, ar^3\).
We need to find how many times larger the fourth term is with respect to the second term. So, we are taking into account \(ar\) and \(ar^3\). How many times... means comparing them to find the ratio. So, \(\frac{ar^3}{ar} = r^2\) represents how many times the fourth term is larger than the second term.
Next step: we know that the sum of first four terms are 5 times the sum of first two terms. We can write:
\(a, ar, ar^2, ar^3 = 5 (a + ar)\)
Arrange it as:
\(a(1 + r + r^2 + r^3) = 5a (r + 1)\)
\(a[1(r^2+1)+r(r^2+1)] = 5a (r + 1)\)
\(a(r+1)(r^2+1) = 5a (r + 1)\)
\(a(r^2+1) = 5a\)
\(r^2 + 1 = 5\)
\(r^2 = 4\)
We need to find how many times larger the fourth term is with respect to the second term. So, we are taking into account \(ar\) and \(ar^3\). How many times... means comparing them to find the ratio. So, \(\frac{ar^3}{ar} = r^2\) represents how many times the fourth term is larger than the second term.
Next step: we know that the sum of first four terms are 5 times the sum of first two terms. We can write:
\(a, ar, ar^2, ar^3 = 5 (a + ar)\)
Arrange it as:
\(a(1 + r + r^2 + r^3) = 5a (r + 1)\)
\(a[1(r^2+1)+r(r^2+1)] = 5a (r + 1)\)
\(a(r+1)(r^2+1) = 5a (r + 1)\)
\(a(r^2+1) = 5a\)
\(r^2 + 1 = 5\)
\(r^2 = 4\)
Re: The sequence of numbers a, ar, ar2, and ar3 are in geometric
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04 Jan 2024, 06:49
04 Jan 2024, 06:49
1
fourth term is how many times larger than the second term , thus ar^3/ar = r^2
so we need to find r^2
Sum of 4 terms = a(1 -r^4)/(1 - r)
Sum of 2 terms = a(1 -r^2)/(1 - r)
thus a(1 -r^4)/(1 - r) = 5a(1 -r^2)/(1 - r)
(1 -r^4) = 5(1 -r^2)
(1 -r^2)(1 +r^2) = 5(1 -r^2)
(1 +r^2) = 5
r^2 = 4
so we need to find r^2
Sum of 4 terms = a(1 -r^4)/(1 - r)
Sum of 2 terms = a(1 -r^2)/(1 - r)
thus a(1 -r^4)/(1 - r) = 5a(1 -r^2)/(1 - r)
(1 -r^4) = 5(1 -r^2)
(1 -r^2)(1 +r^2) = 5(1 -r^2)
(1 +r^2) = 5
r^2 = 4
