Question

If one AM ‘a’ and two GM’s p and q are inserted between any two given numbers then ${p^3} + {q^3} = kapq$, then find k.

Answer 1

Hint: First write the numbers in terms of arithmetic mean. After that use the geometric mean and calculate the value ${p^2}$. Now multiply the term by p to get ${p^3}$. Similarly, calculate for ${q^3}$. Now add both ${p^3}$ and ${q^3}$ simplify it. Then compare the result with the given part and get the value of k.

Formula used:
The arithmetic mean is calculated by,
$AM = \dfrac{{a + b}}{2}$
The geometric mean is calculated by,
$\dfrac{a}{{GM}} = \dfrac{{GM}}{b}$
Cross multiply the value,
$G{M^2} = ab$
where a and b are numbers

Complete step-by-step answer:
Given:- ${p^3} + {q^3} = kapq$ …..(1)
Let the two numbers be x and y.
Then, the arithmetic mean ‘a’ will be,
$a = \dfrac{{x + y}}{2}$
Cross multiply the value to get the equation,
$x + y = 2a$ …..(2)
Now, p and q be the GM between x and y,
Then, x, p, q, y is in GP.
Now, for the geometric mean between x, p, and q, the ratios are equal.
$\dfrac{x}{p} = \dfrac{p}{q}$
Cross multiply the value to get the equation,
${p^2} = xq$
Now multiply both sides by p,
${p^3} = pqx$ ….(3)
Now, for the geometric mean between p, q, and y, the ratios are equal.
$\dfrac{p}{q} = \dfrac{q}{y}$
Cross multiply the value to get the equation,
${q^2} = py$
Now multiply both sides by q,
${q^3} = pqy$ ….(4)
Now add both equations (3) and (4),
${p^3} + {q^3} = pqx + pqy$
Take pq common from the right side of the equation,
${p^3} + {q^3} = pq\left( {x + y} \right)$
Substitute the value of x+y from the equation (2),
${p^3} + {q^3} = pq \times 2a$
Compare the above equation with equation (1),
$kapq = 2apq$
Cancel out common factors from both sides,
$k = 2$

Hence, the value of k is 2.

Note: The arithmetic mean is the simplest and most widely used measure of a mean, or average. It simply involves taking the sum of a group of numbers, then dividing that sum by the count of the numbers used in the series.
The geometric mean is the average rate of return of a set of values calculated using the products of the terms.