Question

Find the G.P in which the 2nd term is √6 and the 6th term is 9√6.

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Answer 1

We know that the formula for the nth term of an G.P is Tn = ar^n-1, where a is the first term, r is the common ratio.

It is given that the 2nd term is T2= √6,  therefore,

⇒ $T_{n} = ar^{2-1}$

⇒ $\sqrt{6}=ar$

⇒  $a = \frac{\sqrt{6} }{r}$------> (1)

It is also given that the 6th term is  $T_{6} = 9\sqrt{6}$ , therefore using equation 1,

⇒ $T_{6} = ar^{n-1}$

⇒ $9\sqrt{6} =\frac{\sqrt{6} }{r} .r^{5}$

⇒  $r^{4} = 9$

⇒ $(r^{2}) ^{2} = (3)^{2}$

⇒ $r^{2} = 3$

⇒ $r = \sqrt{3}$

Now, substitute the value of r in equation 1:

⇒ $a = \frac{\sqrt{6} }{\sqrt{3} } =\sqrt{\frac{6}{3} } =\sqrt{2}$

Therefore, the terms of G.P are:

$a_{1} = \sqrt{2}\\\\a_{2} = a_{1} .r= \sqrt{2} .\sqrt{3} = \sqrt{6} \\\\a_{3} = a_{2} .r= \sqrt{6}.\sqrt{3} = \sqrt{18} = 3\sqrt{2} \\\\a_{4} =a_{3}.r = \sqrt{18} .\sqrt{3} = \sqrt{54} =3\sqrt{6}$

Hence, the G.P is √2, √6, 3√2, 3√6,..and so on.

Hope it helps:)

Answer 2

We know that the formula for the nth term of an G.P is Tn=arn−1, where a is the first term, r is the common ratio.

It is given that the 2nd term is T2=6, therefore,

T2=ar2−1⇒6=ar⇒a=r6__(1)  

It is also given that the 6th term is T6=96, therefore using equation 1,

T6=ar6−1

⇒96=r6×r5

⇒696=r4

⇒r4=9

⇒(r2)2=32

⇒r2=3