SATHEE: Sequences and Series 5 Question 7

Sequences and Series 5 Question 7

7.

If $a, b, c$ are in $A P, a^{2}, b^{2}, c^{2}$ are in HP, then prove that either $a = b = c$ or $a, b, - \frac{c}{2}$ form a GP.

$(2003, 4$ M)

Show Answer

Solution:

Since, $a, b, c$ are in an AP.

$∴ 2 b = a + c$

and $a^{2}, b^{2}, c^{2}$ are in HP.

$\Rightarrow b^{2} = \frac{2 a^{2} c^{2}}{a^{2} + c^{2}} \Rightarrow \frac{a + c}{2} = \frac{2 a^{2} c^{2}}{a^{2} + c^{2}}$

$\Rightarrow (a^{2} + c^{2}) (a^{2} + c^{2} + 2 a c) = 8 a^{2} c^{2}$

$\Rightarrow (a^{2} + c^{2}) + 2 a c (a^{2} + c^{2}) = 8 a^{2} c^{2}$

$\Rightarrow (a^{2} + c^{2}) + 2 a c (a^{2} + c^{2}) + a^{2} c^{2} = 9 a^{2} c^{2}$

$\Rightarrow {(a^{2} + c^{2} + a c)}^{2} = 9 a^{2} c^{2}$

$\Rightarrow a^{2} + c^{2} + a c = 3 a c$

$\Rightarrow 4 b^{2} = - 2 a c \Rightarrow b^{2} = - \frac{a c}{2}$

Hence, $a, b, - \frac{c}{2}$ are in GP.

$∴$ Either $a = b = c$ or $a, b, - \frac{c}{2}$ are in GP.

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