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The sum of two numbers is \(2 \frac{1}{6}\). An even number of arithmetic means are being inserted between them and their sum exceeds their number by 1. Find the number of means inserted.

Find three numbers in G.P. whose sum is 65 and whose product is 3375 .

Three numbers are in G.P. whose sum is 70 . If the extremes be each multiplied by 4 and the mean by 5 , they will be in A.P. Find the numbers.

Find four numbers in G.P. whose sum is 85 and product is 4096 .

In a G.P. the first, third and fifth terms may be considered as the first, fourth and sixteenth terms of an A.P. Determine the fourth term of the A.P., knowing that its first term is 5 and determine \(T_{1}, T_{3}, T_{5}\) of G.P.

The fifth term of a G.P. is 81 whereas its second term is 24 . Find the series and sum of its first eight terms.

In an increasing G.P., the sum of the first and the last term is 66, the product of the second and the last but one term is 128 , and the sum of all the terms is 126 . How many terms are there in the progression?

In a G.P. sum of \(n\) terms is 364 , first term is 1 and the common ratio is 3 . Find \(n\).

A ball is dropped from a height of \(48 \mathrm{ft}\). and rebounds two-third of the distance it falls. If it continues to fall and rebound in this way, how far will it travel before coming to rest?

If \(a, b, c\) are all positive and in H.P., then show that the roots of \(a x^{2}+2 b x+3 c\) are imaginary.