Question

If pth,qth,rth terms of a G.P. are in G.P. then p,q,r are in ?

Answer 1

AP

suppose

pth=a

qth=ar

rth= ar^2

so these are consicutive term

They must be in AP

Follow me and mark as brainlist

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

1.

Three terms x, y, z are said to be in Arithmetic Progression if

$2y = x + z$

2.

Three terms x, y, z are said to be in Geometric Progression if

${y}^{2} = xz$

QUESTION

If pth,qth,rth terms of a G.P. are in G.P. then p,q,r are in

EVALUATION

Let first term = a

Common Ratio = x

So

$p \: term \: = a \: \times {x}^{p - 1}$

$q \: term \: = a \: \times {x}^{q - 1}$

$r \: term \: = a \: \times {x}^{r - 1}$

Now the above terms are in Geometric Progression

So

${( a \: \times {x}^{q - 1}) }^{2} = ( a \: \times {x}^{p - 1}) \times ( a \: \times {x}^{r - 1})$

$\implies \: {a}^{2} \: \times {x}^{2q - 2} = {a}^{2} \times {x}^{p + r - 2}$

$\implies \: 2q - 2 = p + r - 2$

$\implies \: 2q = p + r$

Hence p, q , r are in Arithmetic Progression ( AP)