Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Geometric Mean (G.M.)

If three terms numbers  a, m and  b form a G.P, then  m is said to be the Geometric Mean (G.M) of  a and  b .

Here  \dfrac{m}{a}=\dfrac{b}{m}=r (common ratio)

 \implies m^2=ab\ \implies m=\pm \sqrt{ab}

 m^2=ab\ \implies ab is always positive.

Therefore geometric mean between  a and   b is  \sqrt{ab} or  \sqrt{-ab} .

Geometric Mean of  n terms in G.P

If  a_1,a_2,...a_n are n positive numbers in G.P, then their Geometric Mean is defined as:

 \text{GM} = (a_1.a_2...a_n)^{\frac{1}{n}}=\sqrt[n]{a_1.a_2...a_n}

Note -:

The terms  a_2,a_3,...,a_{n-1} are called the geometric means between  a and  a_n .

 Some Facts about G.P -:

(i) If each term of a GP is multiplied (divided) by a fixed non-zero constant, then the resulting sequence is also a GP.

(ii) If  a_1,a_2,a_3...  b_1,b_2,…b_3 are two geometric progressions, then the sequence  a_1b_1,a_2b_2... is also in GP.

(iii) If we have to take three terms in GP, then we take them as  \dfrac{a}{r},\ a,\ ar and four terms as  \dfrac{a}{r^3},\dfrac{a}{r},\ ar,\ ar^3 .

(iv) If  a_1,a_2,... is a GP  (a_i > 0,\ \forall_i) then  \log\ a_1, \log\ a_2,... is an AP.

Suppose  a_1,a_2,a_3...  is a GP.

Let  a_i=AR^{i-1} , where  A , is the first term and  R is the common ratio of the GP.

Then,  \log\ a_i=\log\ A+(i-1)\log\ R

Then  a_i= \alpha ^{a+(i-1)d} = \alpha ^a(d^d)^{i-1}

Where  \alpha is the base of the logarithm.

This shows that  a_1,a_2,a_3...  is a GP, with first term  \alpha^a and common ratio  \alpha^d .

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