Alan

Here's a counter-example:

A = 13   B = 17   C = 19  

x = 19   y = 38   z = 47

A*x + B*y = 13*19 + 17*38 = 893

C*z = 19*47 = 893

A, B and C are all different primes, so they have no common prime factor.

(x, y and C have common prime factors of course, so perhaps your question was intended to be phrased a little differently).

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If the first term is "a" then the n'th term is a*r^n-1 so  you have:

From term 2: a*3 = 9 so a = 3; and this gives the 12th term as 3*3¹¹ = 531441.

You now have the first and second terms.  I'll leave you to work out the third term.

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I just used Mathcad to tell me Melody!  I'll see if I can derive it.

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Yes; good to get confirmation of the result.  

However, it is probably the steps that are more important than the result for someone wanting to learn how to solve these sorts of equations!

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Here are the steps:

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1/2 + 1/4 is 3/4 so  3/4 of your money = £3.75

This means your money = £3.75*4/3  = £5