Permutations and Combinations

Permutations and Combinations

Question

Find a GP. for which sum of first two terms is –4 and fifth term is 4 times the 3rd term.

Answer

Let the G.P. be a, ar, ar2, ..........
Given          a + ar = -4                                                                        ...(i)
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or                space space space ar to the power of 5 minus 1 end exponent equals space 4. space ar to the power of 3 minus 1 end exponent space rightwards double arrow space ar to the power of 4 minus 4 ar squared space equals space 0 space rightwards double arrow space ar squared left parenthesis straight r squared minus 4 right parenthesis space equals space 0
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rightwards double arrow                         straight r equals plus-or-minus 2.
Putting r = 2 in (i), we get a + 2a = -4 rightwards double arrow a = -4/3.
∴    G.P. is space space space fraction numerator negative 4 over denominator 3 end fraction comma space fraction numerator negative 4 over denominator 3 end fraction left parenthesis 2 right parenthesis. space fraction numerator negative 4 over denominator 3 end fraction left parenthesis 2 right parenthesis squared comma space.......... space or space fraction numerator negative 4 over denominator 3 end fraction comma space fraction numerator negative 8 over denominator 3 end fraction comma space fraction numerator negative 16 over denominator 3 end fraction comma space.........
Now, putting r = -2 in (i), we get a + a(-2) = -4 rightwards double arrow -a = -4 or a = 4.
∴  G.P. is 4, 4(-2), 4(-2)2,   .... or 4, -8, 16, ...........

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