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Three positive numbers form an increasing Geometric Progression. If the middle term is doubled, the new numbers form an Arithmetic Progression. What is the common ratio of the Geometric Progression?
Note: This was an IIT JEE (mains) problem 2014
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1 solution
Let the numbers be a , a r and a r 2 where a is the first term and r be the common ratio of the G.P. When the second term of the G.P. i.e. a r is multiplied by 2, the the new term becomes 2 a r . Now, a , 2 a r and a r 2 are in A.P. In an A.P., the sum of the first and the third term is equal to twice of the second term. Hence, we get: 4 a r = a + a r 2 Taking out a common and furthur cancelling it from both sides,we get: 4 r = 1 + r 2 Solving the above quadratic equation,we get: r = 2 + √ 3
NOTE: We will not take r = 2 − √ 3 as then it will form a decreasing G.P. Hence, the only answer is r = 2 + √ 3 .