Discussion
The first three terms in a G.P. are , \(a\), b and \(c\) where \(a \neq b\) Then the fifth term is
Answer: C
The fifth term is \(\frac{c^{2}}{a}\).
With first term a and ratio r, the third term is \(c = ar^{2}\) and the fifth is \(ar^{4}\).
\[ ar^{4} = \frac{(ar^{2})^{2}}{a} = \frac{c^{2}}{a} \]
Equivalently, the 1st, 3rd and 5th terms are themselves in G.P. with ratio \(r^{2}\), so \(c^{2} = a \times T_{5}\). The options with 2a or 3a in the denominator, or c instead of \(c^{2}\), do not satisfy this; for example with a = 1, r = 2 the terms are 1, 2, 4, 8, 16 and \(\frac{c^{2}}{a} = 16\).
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