First slide

Arithmetic progression

Question

$The numbers 3^{2 \sin 2 α - 1}, 14 and 3^{4 - 2 \sin 2 α} from first three terms of an A.P its 5^{th} term is$

Moderate

A

-25
B

-12
C

40
D

53

Solution

$28 = 3^{2 \sin 2 α - 1} + 3^{4 - 2 \sin 2 α} (\sin c e 2 b = a + c) Let a = 3^{2 \sin 2 α}$
$\begin{matrix} \begin{aligned} 28 = \frac{a}{3} + \frac{3^{4}}{a} \\ \to a^{2} - 84 a + 243 = 0 \to a = 81, 3 \\ If a = 3 \to 3^{2 \sin 2 α} = 3^{1} \\ (i f a = 81 t h e n \sin 2 α = 2 w h i c h i s n o t p o s s i b l e) \\ \to \sin 2 α = 1 / 2 \to 2 α = 30 \\ α = 15^{\circ} \end{aligned} \\ the first three terms are 1, 14, 27 \\ Then t_{5} = a + 4 d \\ = 1 + 52 = 53 \end{matrix}$

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