Question

If a, b and c be in G.P. and $x,y$ be the arithmetic means between a, b and b, c respectively then $\dfrac{a}{x} + \dfrac{c}{y}{\text{ is - }}$
A. 2
B. 1
C. 3
D. 4

Answer 1

Hint-Geometric Progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called common ratio and arithmetic mean is simply average of two numbers. Use these concepts to solve the question.

Complete step-by-step answer:
Given $a,b,c$ are in G.P.
$x = {\text{ arithmetic mean between }}a,b$
$\Rightarrow x = \dfrac{{a + b}}{2} \\\ y = {\text{ arithmetic mean between }}b,c \\\ \Rightarrow y = \dfrac{{b + c}}{2} \\\$
$\Rightarrow \dfrac{a}{x} + \dfrac{c}{y} = \dfrac{a}{{\dfrac{{a + b}}{2}}} + \dfrac{c}{{\dfrac{{b + c}}{2}}} \\\ = \dfrac{{2a}}{{a + b}} + \dfrac{{2c}}{{b + c}} \\\ = \dfrac{{2a(b + c) + 2c(a + b)}}{{(a + b)(b + c)}} \\\ = \dfrac{{2[ab + ac + ac + bc]}}{{ab + ac + {b^2} + bc}}.............(1) \\\ \\\$
Since $a,b,c$ are in a G.P.
$\Rightarrow b = {\text{Geometric mean}} \\\ {\text{b = }}\sqrt {a.c} \\\ {b^2} = ac \\\ {\text{from (1)}} \\\ {\text{ = }}\dfrac{{2[ab + 2ac + bc]}}{{ab + ac + ac + bc}} \\\ = \dfrac{{2[ab + 2ac + bc]}}{{[ab + 2ac + bc]}} = 2 \\\ \\\$
Therefore, the correct option is A.
Note- A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Students must remember the formulas for the sum of n numbers of a G.P. and other common series