Summary

Videos

References

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 â‡’ m^{2} = 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 $\sqrt{\text{ab}}$ = $\sqrt{\text{4 x 16}}$ = $\sqrt{\text{64}}$ = 8

4, 8, 16 form a GP.

In general, many numbers can be inserted between any two given numbers.

Let us find n numbers G_{1}, G_{2}, G_{3},....,G_{n} between two numbers a and b.

Then the sequence a, G_{1}, G_{2}, G_{3},....,G_{n} ,b forms a GP having (n+2) terms.

Let a_{1} = a, a_{2} = G_{1,} a_{3} = G_{2} , ....., a_{n+1}= G_{n}, a_{n+2}= b.

Now, a_{n+2} = b

â‡’ ar^{n+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 G_{1} = t_{2} = ar

= a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$ = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$

G_{2} = t_{3} = ar^{2}

= a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{2}}{\text{n+1}}}$ = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{2}}{\text{n+1}}}$

G_{3} = t_{4} = ar^{3} = a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{3}}{\text{n+1}}}$ = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{3}}{\text{n+1}}}$ â€¦â€¦â€¦.. and so on.

Likewise, we have G_{n} = t_{n+1} = ar^{n} = 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 = $\frac{\text{a + b}}{\text{2}}$ â€“ $\sqrt{\text{ab}}$

= $\frac{\text{a + b - 2}\sqrt{\text{ab}}}{\text{2}}$

= $\frac{{\left(\sqrt{\text{a}}\right)}^{\text{2}}\text{+}{\left(\sqrt{\text{b}}\right)}^{\text{2}}\text{- 2}\sqrt{\text{a}}\text{}\sqrt{\text{b}}}{\text{2}}$

= $$ \frac{{\left(\sqrt{\text{a}}\mathrm{\text{-}}\text{}\sqrt{\text{b}}\right)}^{2}}{2}$$ â‰¥ 0

âˆ´ A â€“ G â‰¥ 0

â‡’ A â‰¥ G

The AM between two positive real numbers is always greater than or equal to their GM.

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 â‡’ m

The Geometric Mean (GM) between a and b is $\sqrt{\text{ab}}$

Ex:

Let a = 4 and b = 16

Then $\sqrt{\text{ab}}$ = $\sqrt{\text{4 x 16}}$ = $\sqrt{\text{64}}$ = 8

4, 8, 16 form a GP.

In general, many numbers can be inserted between any two given numbers.

Let us find n numbers G

Then the sequence a, G

Let a

Now, a

â‡’ ar

ar

r

r = ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$

Then G

= a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$ = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$

G

= a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{2}}{\text{n+1}}}$ = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{2}}{\text{n+1}}}$

G

Likewise, we have G

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 = $\frac{\text{a + b}}{\text{2}}$ â€“ $\sqrt{\text{ab}}$

= $\frac{\text{a + b - 2}\sqrt{\text{ab}}}{\text{2}}$

= $\frac{{\left(\sqrt{\text{a}}\right)}^{\text{2}}\text{+}{\left(\sqrt{\text{b}}\right)}^{\text{2}}\text{- 2}\sqrt{\text{a}}\text{}\sqrt{\text{b}}}{\text{2}}$

= $$ \frac{{\left(\sqrt{\text{a}}\mathrm{\text{-}}\text{}\sqrt{\text{b}}\right)}^{2}}{2}$$ â‰¥ 0

âˆ´ A â€“ G â‰¥ 0

â‡’ A â‰¥ G

The AM between two positive real numbers is always greater than or equal to their GM.

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 â‡’ m^{2} = 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 $\sqrt{\text{ab}}$ = $\sqrt{\text{4 x 16}}$ = $\sqrt{\text{64}}$ = 8

4, 8, 16 form a GP.

In general, many numbers can be inserted between any two given numbers.

Let us find n numbers G_{1}, G_{2}, G_{3},....,G_{n} between two numbers a and b.

Then the sequence a, G_{1}, G_{2}, G_{3},....,G_{n} ,b forms a GP having (n+2) terms.

Let a_{1} = a, a_{2} = G_{1,} a_{3} = G_{2} , ....., a_{n+1}= G_{n}, a_{n+2}= b.

Now, a_{n+2} = b

â‡’ ar^{n+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 G_{1} = t_{2} = ar

= a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$ = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$

G_{2} = t_{3} = ar^{2}

= a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{2}}{\text{n+1}}}$ = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{2}}{\text{n+1}}}$

G_{3} = t_{4} = ar^{3} = a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{3}}{\text{n+1}}}$ = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{3}}{\text{n+1}}}$ â€¦â€¦â€¦.. and so on.

Likewise, we have G_{n} = t_{n+1} = ar^{n} = 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 = $\frac{\text{a + b}}{\text{2}}$ â€“ $\sqrt{\text{ab}}$

= $\frac{\text{a + b - 2}\sqrt{\text{ab}}}{\text{2}}$

= $\frac{{\left(\sqrt{\text{a}}\right)}^{\text{2}}\text{+}{\left(\sqrt{\text{b}}\right)}^{\text{2}}\text{- 2}\sqrt{\text{a}}\text{}\sqrt{\text{b}}}{\text{2}}$

= $$ \frac{{\left(\sqrt{\text{a}}\mathrm{\text{-}}\text{}\sqrt{\text{b}}\right)}^{2}}{2}$$ â‰¥ 0

âˆ´ A â€“ G â‰¥ 0

â‡’ A â‰¥ G

The AM between two positive real numbers is always greater than or equal to their GM.

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 â‡’ m

The Geometric Mean (GM) between a and b is $\sqrt{\text{ab}}$

Ex:

Let a = 4 and b = 16

Then $\sqrt{\text{ab}}$ = $\sqrt{\text{4 x 16}}$ = $\sqrt{\text{64}}$ = 8

4, 8, 16 form a GP.

In general, many numbers can be inserted between any two given numbers.

Let us find n numbers G

Then the sequence a, G

Let a

Now, a

â‡’ ar

ar

r

r = ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$

Then G

= a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$ = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{1}}{\text{n+1}}}$

G

= a x ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{2}}{\text{n+1}}}$ = a ${\left(\frac{\text{b}}{\text{a}}\right)}^{\frac{\text{2}}{\text{n+1}}}$

G

Likewise, we have G

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 = $\frac{\text{a + b}}{\text{2}}$ â€“ $\sqrt{\text{ab}}$

= $\frac{\text{a + b - 2}\sqrt{\text{ab}}}{\text{2}}$

= $\frac{{\left(\sqrt{\text{a}}\right)}^{\text{2}}\text{+}{\left(\sqrt{\text{b}}\right)}^{\text{2}}\text{- 2}\sqrt{\text{a}}\text{}\sqrt{\text{b}}}{\text{2}}$

= $$ \frac{{\left(\sqrt{\text{a}}\mathrm{\text{-}}\text{}\sqrt{\text{b}}\right)}^{2}}{2}$$ â‰¥ 0

âˆ´ A â€“ G â‰¥ 0

â‡’ A â‰¥ G

The AM between two positive real numbers is always greater than or equal to their GM.