Question

If ' n ' geometric means are inserted between 16/27 and 243/16 such that the ratio of

(n-1)th mean and 4th mean is

9 : 4, find the value of n.

Answer 1

The value of n is 7.

• A series in mathematics is essentially a description of the process of successively adding an infinite number of quantities to a specified initial quantity. A significant component of calculus and its generalisation, mathematical analysis, is the study of series.
• Most branches of mathematics use series, including combinatorics, where generating functions are used to explore finite structures. In addition to being widely utilised in mathematics, infinite series are also used extensively in physics, computer science, statistics, and finance, among other quantitative fields.

Here, according to the given information, we are given that,

Let there be n geometric means that are inserted between $\frac{16}{27}$ and $\frac{243}{16}$.

Now, the first term of the series must be $\frac{16}{27}$.

Then, a that is the first term is $\frac{16}{27}$.

Then, the $(n+2)$th term must be $\frac{243}{16}$.

Let the common ratio be assumed to be r.

Then by the rules of geometric mean, we have that,

$\frac{243}{16} = \frac{16}{27}.r^{n+1}$

Or, $r^{n+1}= \frac{3}{2} ^{8}$

Now, we can say that, the fourth mean is equal to the fifth term of the series and that is equal to $ar^{4}$.

Then, we get, (n-1)th mean is equal to $ar^{n-1}$.

Then, we get,

$\frac{ar^{n-1}}{ar^{4}} = \frac{9}{4}$

Or, n + 1 = 4n - 20.

Or, 3n = 21

Or, n = 7.

Hence, the value of n is 7.

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