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
$\therefore$ T₄ =81 = 3r^4-1 $\Rightarrow$ r³ = $ \frac { 81 } { 3 }$
$\Rightarrow$ r³= 27 $\Rightarrow$ r³ = 3³
On comparing the base of power 3 from both sides, we get
r = 3
$\therefore$ a = Ar = 3 $\times$ 3 = 9
and b = Ar² = 3 $\times$ 3² = 27

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