Permutations and Combinations

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.

Answer

Let a, ar, ar2 be three numbers in G.P.
Since their sum is 56
∴           a + ar + ar2 = 56
rightwards double arrow          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.
rightwards double arrow space space space space space space space space left parenthesis ar minus 7 right parenthesis space minus space left parenthesis straight a minus 1 right parenthesis space equals space left parenthesis ar squared minus 21 right parenthesis space minus space left parenthesis ar minus 7 right parenthesis   rightwards double arrow space space space ar minus 7 minus straight a plus 1 space equals space ar squared space minus 21 minus ar plus 7
rightwards double arrow            ar squared minus 2 ar plus straight a equals 8                                 rightwards double arrow  straight a left parenthesis straight r squared minus 2 straight r plus 1 right parenthesis space equals space 8                  ...(ii)
Dividing (i) by (ii), we get fraction numerator straight a left parenthesis straight r squared plus straight r plus 1 right parenthesis over denominator straight a left parenthesis straight r squared minus 2 straight r plus 1 right parenthesis end fraction space equals space 56 over 8
rightwards double arrow          straight r squared plus straight r plus 1 space equals space 7 left parenthesis straight r squared minus 2 straight r plus 1 right parenthesis space rightwards double arrow space space 6 straight r squared minus 15 straight r plus 6 space equals space 0 space space rightwards double arrow space 2 straight r squared minus 5 straight r plus 2 space equals space 0
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when                    r = 2, from (i), space space space straight a left parenthesis 1 plus 2 plus 2 squared right parenthesis space equals space 56 space rightwards double arrow space space 7 straight a space equals space 56 space space rightwards double arrow space space straight a space equals space 8
∴ Numbers are 8, 8(2), 8(2)2 or 8, 16, 32
When                           straight r equals space 1 half from (i), straight a open parentheses 1 plus 1 half plus 1 fourth close parentheses space equals space 56 space space space rightwards double arrow space space straight a open parentheses 7 over 4 close parentheses space equals space 56 space space space space rightwards double arrow space space straight a space equals space 32
∴    Numbers are    32 comma space 32 open parentheses 1 half close parentheses comma space 32 open parentheses 1 half close parentheses squared  or 32, 16, 8

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