Insert two numbers between 3 and 81, so that resulting sequence i… - Vidyadip

Question

Insert two numbers between $3$ and $81,$ so that resulting sequence is GP.

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Answer

Let the two numbers be a and b, then $3, a, b, 81$ are in GP.
$\because n^{th}$^ term, $T_n = Ar^{n-1}$
$\therefore T_4 =81 = 3r^{4-1} \Rightarrow r^3 = \frac { 81 } { 3 }$
$\Rightarrow r^{3}= 27 \Rightarrow r^3 = 3^3$
On comparing the base of power 3 from both sides, we get
$r = 3$
$\therefore a = Ar = 3 \times 3 = 9$
and $b = Ar^2 = 3 \times 3^2 = 27$

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