First slide

Definitions of sequence; series and progressions

Question

Suppose a, b, c are distinct real numbers. If $a, b, c$ are in A.P. and $a^{2}, b^{2}, c^{2}$ are in H.P., then

Moderate

A

$- \frac{a}{2}, b, c$ are in G.P.
B

$a + b = c$
C

$a = b + c$
D

$a, b, c$ are in G.P.

Solution

$\begin{aligned} 2 b = a + c, b^{2} = \frac{2 a^{2} c^{2}}{a^{2} + c^{2}} \\ \Rightarrow \frac{(a + c)^{2}}{4} = \frac{2 a^{2} c^{2}}{a^{2} + c^{2}} \end{aligned}$
$\begin{array}{l} \Rightarrow {(a^{2} + c^{2})}^{2} + 2 a c (a^{2} + c^{2}) - 8 a^{2} c^{2} = 0 \\ \Rightarrow (a^{2} + c^{2} + 4 a c) (a^{2} + c^{2} - 2 a c) = 0 \\ \Rightarrow [(a + c)^{2} + 2 a c] (a - c)^{2} = 0 \\ \Rightarrow (a + c)^{2} + 2 a c = 0 [∵ a \neq c] \\ \Rightarrow 4 b^{2} + 2 a c = 0 \end{array}$
$\Rightarrow - \frac{a}{2}, b, c$ are in G.P.

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