Question

If \[a,b,c\] be in A.P. and \[b,c,d\] be in H.P., then
A. \[ab = cd\]
В. \[ad = bc\]
C. \[ac = bd\]
D. \[abcd = 1\]

Answer 1

Hint:
Some values' means are the average of those values. The geometric mean of \[n\] values is calculated by taking the \[{n^{th}}\] root of the product of those n values. The harmonic mean of some values is a mean that is the reciprocal of the arithmetic mean of those values' reciprocals.
Formula used:
The A.M of a,b term of A.P is \[\frac{{(a + b)}}{2}\] )
The G.M of a,b term of G.P is \[\sqrt {ab} \]
The H.M of a, b term of H.P is \[\frac{{2ab}}{{(a + b)}}\]
Complete step-by-step solution:
We've been told that “a”, “b”, and “c” are in AP, “b”, “c”, and “d” are in HP.
We are to determine the sequence relationship between the terms a, b, c and d.
Since, \[a,b,c\] are in A.P.
In AP, the arithmetic mean of the three terms a, b, and c is given as twice the middle is equal to the sum of the first and third terms.
Therefore by using the formula for Arithmetic mean, we have
\[b = \frac{{a + c}}{2}\] --- (1)
Since, \[b,c,d\] are in H.P.
Therefore, by using the formula for Harmonic mean, we have
\[c = \frac{{2bd}}{{b + d}}\] --- (2)
Now, we have to multiply the equation (1) and (2), we obtain
\[bc = \frac{{a + c}}{2} \times \frac{{2bd}}{{b + d}}\]--- (3)
Multiply both sides of the equation (3) by \[b + d\].
\[{b^2}c + bcd = abd + bcd\]
Now, we have to flip the above equation for easy simplification:
\[abd + bcd = {b^2}c + bcd\]
Now, add \[ - bcd\] to both sides of the above equation.
\[abd + bcd + (- bcd) = {b^2}c + bcd + ( - bcd) = {b^2}c\]
We have to divide both sides of the above equation by \[bd\]:
\[\frac{{abd}}{{bd}} = \frac{{{b^2}c}}{{bd}}\]--- (4)
\[\frac{{abd}}{{bd}} = \frac{{{b^2}c}}{{bd}}\]
Simplify the above equation by canceling similar terms:
\[a = \frac{{bc}}{d}\]
This can be written as,
\[ad = bc\]
The above relation clearly denotes the geometric mean of the terms a, b, c and d in the given order.
Therefore, if \[a,b,c\] be in A.P. and \[b,c,d\] be in H.P., then \[ad = bc\]
Hence, the option B is correct.
Note:
To answer the above question, you must remember the formulas for the three means of different sequences. Try not to remove any of the terms a, b, c, or d from the given relations because we need to find the relationship between these terms. Simply assigning some small integral values to the terms a, b, c and d will allow you to check the answer.