Permutations and Combinations

Question

# The sum of three numbers in GP. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the numbers.

Let a, ar, ar2 be three numbers in G.P.
Since their sum is 56
∴           a + ar + ar2 = 56
a(1 + r + r2) = 56                                                                                   ...(i)

It is given that if we subtract 1, 7, 21 from the numbers respectively then the resulting numbers form an A.P.
∴    a - 1, ar - 7, ar- 21 are in A.P.

...(ii)
Dividing (i) by (ii), we get

when                    r = 2, from (i),
∴ Numbers are 8, 8(2), 8(2)2 or 8, 16, 32
When                            from (i),
∴    Numbers are      or 32, 16, 8