Question

The second term of a G.P is 12 more than the first term, given that the common ratio is half of the first term, find the third term of the G.P

Answer 1

Given:-

• The second term of a G.P is 12 more than the first term.
• the common ratio is half of the first term.

To Find:-

• the third term of the G.P.

Solution:-

Let a be the first term

a₂ be the 2nd term

a₃ be the 3rd term

and

r be the common ratio

It is given that The second term of a G.P is 12 more than the first term.

So,

$a₂ = a + 12$

and,

The common ratio is half of the first term.

$r = \frac{a}{2}$

We Know that

In a G.P ,If we have to Find the Common Ratio (r) Then,

divide the second term of the sequence with the first term or simply find the ratio of any two consecutive terms by taking the previous term in the denominator.

That Means,

$r= \frac{ a₂}{a} = \frac{a₃}{ a₂ }$

Or

$= > r= \frac{ a₂}{a}$

$= > ar = a₂$

Now , Substituting the value of ar and a₂ in this equation

$= > a + 12 = a \times \frac{a}{2}$

$= > a + 12 = \frac{ {a}^{2} }{2}$

$= > {a}^{2} = 2(a + 12)$

$= > a² - 2a - 24 = 0$

This is a Quadratic equation

So we will Find Zeros of this equation by using Factorisation Method.

Now, Splitting the middle term

$= > a² - 6a + 4a - 24 = 0$

$= > a(a - 6) + 4(a - 6) = 0$

$= > (a - 6) (a + 4) = 0$

$a = 6 \: or \: a = - 4$

So, there are 2 possible values of a.

Now, Putting the both values of a to find a₃

$\: \huge\bf\underline \red{\underline{a₃ = ar²}}$

If a=6:

Then

$r = \frac{a}{2} = \frac{6}{2} = 3$

So,

$a₃ = 6 \times {3}^{2} = 6 \times 9$

$= > 54$

And

If a=-4:

Then

$r = \frac{ - 4}{2} = - 2$

And

$a₃ = - 4 \times { - 2}^{2} = - 4 \times 4$

$= > - 16$

∴third term can be either 54 or -16.