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The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers. |
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Answer» Let the three numbers be \(\frac{a}{r}\),a, ar. ∴ According to the question ⇒ \(\frac{a}{r} \times a \times ar\) = 216 .......(1) ⇒ a3 = 216 ⇒ a = 6 2,8,6 is added to them \(\therefore\) \(\frac{a}{r} + 2\), a + 8, ar + 6 The above sequence is in AP. We know in AP. 2b = a + c ⇒ 2(a + 8) = \(\frac{a}{r}\) + 2 + ar + 6 Substituting a = 6 in above equation we get, ⇒ 2(6 + 8) = \(\frac{6}{r}\) + 2 + 6r + 6 ⇒ 28 = \(\frac{6 + 6r^2 + 8r}{r}\) ⇒ 28r = 6 + 6r2 + 8r ⇒ 6r2 - 18r - 2r - 6 = 0 ⇒ 6r(r-3) - 2(r - 3) = 0 ⇒ r = 3 or r = 1/3 ∴ Now the equation will be ⇒ \(\frac{6}{3}, 6,6 \times 3\) or \(\frac{6}{\frac{1}{2}}, 6 ,6 \times \frac{1}{3}\) ⇒ 2,6,18 or 18,6,2 ∴ The three numbers are 2,6,18. |
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