finding the terms of a geometric progression

Hi,if we are given two equidistant terms of a geometric progression and also the sum of its terms , then how can we find the actual terms of the G.P? e.g if we are given the 3rd and last terms of the G.P as 12 and 48 and the sum of terms is 393 then the number of terms is 7 and they are: 3,6,12,24,48,96,192 .

I am aware of the following relations:

1. The sum of terms of G.P = a(r^n-1)/(r-1)

2. Product of equidistant terms of a G.P = product of extremes.

But am not sure how to apply them here to get the result.

Thanks.

Re: finding the terms of a geometric progression

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Originally Posted by

**pranay** if we are given the 3rd and last terms of the G.P as 12 and 48 and the sum of terms is 393 then the number of terms is 7 and they are: 3,6,12,24,48,96,192 .

How are 12 and 48 the "3rd and last" terms of the sequence? Isn't 192 the last term?

I am not sure what is given in your problem. I assume you are given that 12 are 48 are terms equidistant to the extremes and that the sum of terms is 393. Are you given the sequence length and the fact that 12 is the third term?

Re: finding the terms of a geometric progression

Can you make us a clear question here. Those terms add up to 381.

Re: Finding the terms of a geometric progression

Quote:

Originally Posted by

**pranay** Hi,if we are given two equidistant terms of a geometric progression and also the sum of its terms , then how can we find the actual terms of the G.P? e.g if we are given the 3rd and last terms of the G.P as 12 and 48 and the sum of terms is 393 then the number of terms is 7 and they are: 3,6,12,24,48,96,192 .

I think you are talking about the middle term of a GP with an odd number of terms. (The middle term is equidistant from the first and last terms (with respect to position in the progression) but we don't call it the "equidistant term"; that's bad English. (Shake)) By the way, if the middle term of a progression is the 3rd term, then the progression has 5 terms, not 7. Your GP is simply $\displaystyle 3,6,12,24,48$ (and the sum of terms is $\displaystyle 93$ (not 393)).

Quote:

Originally Posted by

**pranay** 2. Product of equidistant terms of a G.P = product of extremes.

Better way to put it: Square of middle term of a GP = product of first and last terms.

Right, now that we've cleared things up, we can answer your question. The 3rd and 5th terms of a GP are $\displaystyle 12$ and $\displaystyle 48$ respectively, and the sum of the 5 terms is $\displaystyle 93.$ Find the GP.

__Solution:__

Let the first term be $\displaystyle a.$ Using the fact that the square of the middle term is equal to the product of first and last terms, $\displaystyle 12^2=144=48a$ $\displaystyle \implies$ $\displaystyle a=3.$ To find the common ratio, let it be $\displaystyle r.$ Equating the 3rd term from formula, $\displaystyle 3r^2=12$ $\displaystyle \implies$ $\displaystyle r=\pm2.$ But if $\displaystyle r=-2$ then, using the sum-of-terms formula, the sum of the GP would be $\displaystyle 33$ and not $\displaystyle 93.$ Hence $\displaystyle r=2$ and the GP is $\displaystyle 3,6,12,24,48.$

Re: finding the terms of a geometric progression

Quote:

Originally Posted by

**emakarov** How are 12 and 48 the "3rd and last" terms of the sequence? Isn't 192 the last term?

I am not sure what is given in your problem. I assume you are given that 12 are 48 are terms equidistant to the extremes and that the sum of terms is 393. Are you given the sequence length and the fact that 12 is the third term?

i am extremely sorry ..given are the 3rd and *3rd* last terms of the g.p(equidistant from extremes) .Here 12 and 48 . and the sum of terms is 381.