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Let a_1, a_2, a_3, ... be an arithmetic sequence.  If a_5/a_3 = 3, then find a_4/a_2.

 Dec 21, 2020
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If the first term is a and the common difference d, the progression will be

a, a + d, a + 2d, a + 3d, a + 4d, ... so 

\(\displaystyle \frac{a_{5}}{a_{3}}=\frac{a+4d}{a+2d}=3\\ \text{so} \\a+4d=3a+6d, \\2a+2d=0,\\d=-a.\)

 

The progression will be a, 0, -a, -2a, -3a, ... in which case 

\(\displaystyle \frac{a_{4}}{a_{2}}=\frac{-2a}{0},\)

which you might take to be meaningless (since your not allowed to divide by zero), or infinite ?

 Dec 22, 2020

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