Question

3 numbers form an increasing GP . if the middle term is doubled, then the new numbers are in AP. find the common ratio of the GP

Answer 1

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Answer 2

♧♧HERE IS YOUR ANSWER♧♧

Let, the GP is

a, ar and ar²,

where a is the first term of the GP and r is the common ratio.

When, the middle term is double is doubled, we get an AP :

a, 2ar, ar²

So, the condition of AP suggests :

2nd term - 1st term = 3rd term - 2nd term

=> 2ar - a = ar² - 2ar

=> 2r - 1 = r² - 2r [eliminating a]

=> r² - 4r + 1 = 0

Using Sreedhar Acharya's method :

r = [- (-4) ± √{ (-4)² - 4.1.1 }]/(2.1)

= [4 ± √(16 - 4)]/2

= [4 ± √12]/2

= [4 ± 2√3]/2

= (2 ±√3)

So, the common ratio of the GP is (2 ±√3).

♧♧HOPE THIS HELPS♧♧