Solution attached.
Edit 13/04/'19:
Mahmoudfawzy83 asked me to rewrite the solution as forum post, which I pursued and illustrated in the .pdf attached.
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:
\(a(1 + r + r^2 + r^3) = 5a (r + 1)\)
\(a[1(r^2+1)+r(r^2+1)] = 5a (r + 1)\)
\(a(r+1)(r^2+1) = 5a (r + 1)\)
\(a(r^2+1) = 5a\)
\(r^2 + 1 = 5\)
\(r^2 = 4\)
Hope it's clear.
Attachments
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