Geometric Mean In GP - Practice Questions & MCQ

Updated on Sep 18, 2023 18:34 AM

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Geometric Mean is considered one of the most asked concept.
16 Questions around this concept.

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Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}$

if $\frac{1}{M}:G$ is 4:5 then a:b can be:

$1:4$

$1:2$

$2:3$

$3:4$

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For any three positive real numbers a, b and c, 9(25a²+b²)+25(c²−3ac)=15b(3a+c). Then:

b, c and a are in A.P.

a, b and c are in A.P.

a, b and c are in G.P.

b, c and a are in G.P.

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Concepts Covered - 2

Geometric Mean

If three terms are in G.P., then the middle term is called the Geometric Mean (G.M.) of the other two numbers. So if, a, b and c are in G.P., then b is GM of a and c,

If $a_1,a_2,a_3,.....,a_n$ are n positive numbers, then the Geometric Mean of these numbers is given by $G=\sqrt[n]{a_1\cdot a_2\cdot a_3\cdot.....\cdot a_n} .$

If a and b are two numbers and G is the GM of a and b. Then, a, G, b are in geometric progression.

Hence, $G = \sqrt{a\cdot b}$ .

Insertion of n-Geometric Mean Between a and b

Let $\\\mathrm{\;G_1,G_2,G_3....,G_n}$ be n geometric mean between two numbers a and b. Then, $\mathrm{\mathit{a},G_1,G_2,G_3....,G_n,\mathit{b}}$ is an G.P. Clearly, this G.P. contains n + 2 terms.

$\\\mathrm{now,\;b=(n+2)^{th}\;term=ar^{n+2-1}}\\\mathrm{\therefore r=\left ( \frac{b}{a} \right )^{\frac{1}{n+1}}}\\\mathrm{[where,\;r=common\;ratio]}\\\mathrm{\therefore G_1=ar,\;G_2=ar^2,\;G_3=ar^3,.....,Ga_n=ar^n}\\\mathrm{\Rightarrow G_1=a\left ( \frac{b}{a} \right )^{\frac{1}{n+1}},\;G_2=a\left ( \frac{b}{a} \right )^{\frac{2}{n+1}},\;G_3=a\left ( \frac{b}{a} \right )^{\frac{3}{n+1}}......}\\\mathrm{G_n=a\left ( \frac{b}{a} \right )^{\frac{n}{n+1}}}$

Important Property of GM

Important Property of GM

The product of n geometric mean between a and b is equal to the n^th power of a single geometric mean between a and b.

If a and b are two numbers and $\\\mathrm{\;G_1,G_2,G_3....,G_n}$ are n-geometric mean between a and b, then $a,\mathrm{\;G_1,G_2,G_3....,G_n},b$ will be in geometric progression.

So, Product of n-G.M’s between a and b is

$\\\mathrm{G_1\cdot G_2\cdot G_3\cdot....\cdot G_n}=(ar)(ar^2)(ar^3)....(ar^n)\\\\\Rightarrow \left ( a^{1+1+1+...\mathrm{n-times}} \right )\left ( r^{1+2+3+...+n} \right )\\\Rightarrow a^n\left ( r^{\left ( \frac{n(n+1)}{2} \right )} \right )\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1+2+.....+n = \frac{n(n+1)}{2} \,\,Using\,\, sum\,\, of\,\, AP)\\\mathrm{replace\;\mathit{r}\;with\;\left ( \frac{b}{a} \right )^{\left ( \frac{1}{n+1} \right )}}\\\Rightarrow a^n\cdot\left [ \left ( \frac{b}{a} \right )^{\frac{1}{n+1}} \right ]^{\frac{n(n+1)}{2}}=a^n\left ( \frac{b}{a} \right )^{\frac{n}{2}}\\\Rightarrow \left ( a \right )^{\frac{n}{2}}\left ( b \right )^{\frac{n}{2}}=\left ( \sqrt{a\cdot b} \right )^n\\\mathrm{=[G.M.\;of\;\mathit{a}\;and\;\mathit{b}]^n}$

Geometric Mean In GP - Practice Questions & MCQ

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Concepts Covered - 2

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