Permutations and Combinations

Permutations and Combinations

Question

Find four numbers forming a GP. in which the third term is greater than the first by 9 and the second term is greater than the fourth by 18.

Answer

Let a, ar, ar2, ar3 be the four numbers in G.P.
Since the third number is greater than the first by 9
∴                         ar2 = a + 9         rightwards double arrow a(r2 - 1) = 9                           ...(i)
Since the second number is greater than the fourth by 18
∴                          ar = ar3 + 18      rightwards double arrow ar(r2 - 1)  = -18                      ...(ii)
Dividing (ii) by (i), we get
                fraction numerator ar left parenthesis straight r squared minus 1 right parenthesis over denominator straight a left parenthesis straight r squared minus 1 right parenthesis end fraction space equals space fraction numerator negative 18 over denominator 9 end fraction space space space space space space space space space space space space rightwards double arrow space space straight r space equals space minus 2
Using r = -2 in (i), we get
            straight a left square bracket left parenthesis negative 2 right parenthesis squared minus 1 right square bracket space equals space 9     rightwards double arrow       a(4 - 1) = 9 rightwards double arrow           a = 3
∴    Numbers are 3, 3(-2), 3(-2)2, 3(-2)3  or 3, -6, 12, -24

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