Notes On Geometric Mean - CBSE Class 11 Maths
The geometric mean (GM) between two numbers is a number, which, when placed between them, forms with them a geometric progression.   Formula to calculate the geometric mean   Let m be the geometric mean between two numbers a and b, then a, m, b are in GP. The common ratio is m/a or b/m, such that m/a =b/m ⇒ m2 = ab (m = $\sqrt{\text{ab}}$). The Geometric Mean (GM) between a and b is $\sqrt{\text{ab}}$   Ex: Let a = 4 and b = 16 Then $\sqrt{\text{ab}}$  =  $\sqrt{\text{4 x 16}}$   =  = 8 4, 8, 16 form a GP.   In general, many numbers can be inserted between any two given numbers.   Let us find n numbers G1, G2, G3,....,Gn between two numbers a and b.   Then the sequence a, G1, G2, G3,....,Gn ,b forms a GP having (n+2) terms. Let a1 = a, a2 = G1, a3 = G2 , .....,  an+1= Gn, an+2= b. Now, an+2 = b ⇒ arn+2-1 = b, where r is the common ratio of this GP     ar(n+1) = b     r(n+1) = b/a       r = ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$   Then  G1 = t2 = ar       = a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$   = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$              G2 = t3 = ar2       = a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{2}}{\text{n+1}}}$   = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{2}}{\text{n+1}}}$           G3 = t4 = ar3 = a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{3}}{\text{n+1}}}$   = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{3}}{\text{n+1}}}$     ……….. and so on. Likewise, we have Gn = tn+1 = arn = a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{n}}{\text{n+1}}}$   = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{n}}{\text{n+1}}}$      Relationship between AM and GM The arithmetic mean between two numbers is their average. The arithmetic mean (AM) between two numbers is a number, which, when placed between them, forms with them an arithmetic progression.   Let A and G be the AM and GM of two positive real numbers a and b, respectively. Then A = $\frac{\text{a + b}}{\text{2}}$  and = $\sqrt{\text{ab}}$ . Now, A - G =  $\frac{\text{a + b}}{\text{2}}$  – $\sqrt{\text{ab}}$                  = $\frac{\text{a + b - 2}\sqrt{\text{ab}}}{\text{2}}$                 = $\frac{{\left(\sqrt{\text{a}}\right)}^{\text{2}}\text{+}{\left(\sqrt{\text{b}}\right)}^{\text{2}}\text{- 2}\sqrt{\text{a}}\text{}\sqrt{\text{b}}}{\text{2}}$                   = $\frac{{\left(\sqrt{\text{a}}\mathrm{\text{-}}\text{}\sqrt{\text{b}}\right)}^{2}}{2}$ ≥ 0 ∴  A – G ≥ 0 ⇒ A ≥ G The AM between two positive real numbers is always greater than or equal to their GM.

#### Summary

The geometric mean (GM) between two numbers is a number, which, when placed between them, forms with them a geometric progression.   Formula to calculate the geometric mean   Let m be the geometric mean between two numbers a and b, then a, m, b are in GP. The common ratio is m/a or b/m, such that m/a =b/m ⇒ m2 = ab (m = $\sqrt{\text{ab}}$). The Geometric Mean (GM) between a and b is $\sqrt{\text{ab}}$   Ex: Let a = 4 and b = 16 Then $\sqrt{\text{ab}}$  =  $\sqrt{\text{4 x 16}}$   =  = 8 4, 8, 16 form a GP.   In general, many numbers can be inserted between any two given numbers.   Let us find n numbers G1, G2, G3,....,Gn between two numbers a and b.   Then the sequence a, G1, G2, G3,....,Gn ,b forms a GP having (n+2) terms. Let a1 = a, a2 = G1, a3 = G2 , .....,  an+1= Gn, an+2= b. Now, an+2 = b ⇒ arn+2-1 = b, where r is the common ratio of this GP     ar(n+1) = b     r(n+1) = b/a       r = ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$   Then  G1 = t2 = ar       = a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$   = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$              G2 = t3 = ar2       = a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{2}}{\text{n+1}}}$   = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{2}}{\text{n+1}}}$           G3 = t4 = ar3 = a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{3}}{\text{n+1}}}$   = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{3}}{\text{n+1}}}$     ……….. and so on. Likewise, we have Gn = tn+1 = arn = a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{n}}{\text{n+1}}}$   = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{n}}{\text{n+1}}}$      Relationship between AM and GM The arithmetic mean between two numbers is their average. The arithmetic mean (AM) between two numbers is a number, which, when placed between them, forms with them an arithmetic progression.   Let A and G be the AM and GM of two positive real numbers a and b, respectively. Then A = $\frac{\text{a + b}}{\text{2}}$  and = $\sqrt{\text{ab}}$ . Now, A - G =  $\frac{\text{a + b}}{\text{2}}$  – $\sqrt{\text{ab}}$                  = $\frac{\text{a + b - 2}\sqrt{\text{ab}}}{\text{2}}$                 = $\frac{{\left(\sqrt{\text{a}}\right)}^{\text{2}}\text{+}{\left(\sqrt{\text{b}}\right)}^{\text{2}}\text{- 2}\sqrt{\text{a}}\text{}\sqrt{\text{b}}}{\text{2}}$                   = $\frac{{\left(\sqrt{\text{a}}\mathrm{\text{-}}\text{}\sqrt{\text{b}}\right)}^{2}}{2}$ ≥ 0 ∴  A – G ≥ 0 ⇒ A ≥ G The AM between two positive real numbers is always greater than or equal to their GM.

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