Name: Finding N Arithmetic Means Between Two Numbers
Uploaded: 2019-07-19
Duration: 3 min 37 s
Description: When three numbers are in A.P., the middle term is called the arithmetic mean of the other two numbers. Watch this video for complete information.

Question

Insert

5

geometric means between

576

and

9

. Show that their product is equal to the fifth power of the single

G . M .

between the given numbers.

Accepted Answer

Let $G_{1}, G_{2}, G_{3}, G_{4}$ and $G_{5}$ be five numbers between $576$ and $9$ so that $576, G_{1}, G_{2}, G_{3}, G_{4}, G_{5}, 9$ are in $G . P .$

We know that, the general form of a $G . P .$ is $a, ar, {ar}^{2}, {ar}^{3}, a r^{4}, a r^{5}, a r^{6}$ and so on.

$∴$ $9 = a_{7} = a r^{6} = 576 \times r^{6}$

$\Rightarrow r^{6} = \frac{9}{576} = \frac{1}{64} = \frac{1}{2^{6}} = {(\frac{1}{2})}^{6}$

$\Rightarrow r = \frac{1}{2}$

where $r$ be the common ratio of $G . P .$

Thus,

$G_{1} = a r = 576 \times \frac{1}{2} = 288$ ,

$G_{2} = a_{3} = a r^{2} = 576 \times {(\frac{1}{2})}^{2} = 144$ ,

$G_{3} = a_{4} = a r^{3} = 576 \times {(\frac{1}{2})}^{3} = 72$ ,

$G_{4} = a_{5} = a r^{4} = 576 \times {(\frac{1}{2})}^{4} = 36$ and

$G_{5} = a_{6} = a r^{5} = 576 \times {(\frac{1}{2})}^{5} = 18$ .

Hence, the required terms of $G . P .$ are $288, 144, 72, 36, 18$

Let $G$ be the $G . M .$ of $576$ and $9$

We know that, geometric mean of $a, b$ $= \sqrt{a b}$

$\Rightarrow G = \sqrt{576 \times 9} = 24 \times 3 = 72$

Now, $G_{1} \cdot G_{2} \cdot G_{3} \cdot G_{4} \cdot G_{5} = 288 \times 144 \times 72 \times 36 \times 18$

$\Rightarrow (72 \times 4) \times (72 \times 2) \times 72 \times 36 \times 18$

$\Rightarrow 72 \times 72 \times 72 \times (36 \times 2) \times (18 \times 4)$

$\Rightarrow 72 \times 72 \times 72 \times 72 \times 72$

$\Rightarrow (72)^{5} = G^{5}$

Hence, their product is equal to the fifth power of single $G . M .$ between the given numbers.

Insert $5$ geometric means between $576$ and $9$ . Show that their product is equal to the fifth power of the single $G . M .$ between the given numbers.

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

Insert 5 geometric means between 576 and 9. Show that their product is equal to the fifth power of the single G.M. between the given numbers.

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

Insert $5$ geometric means between $576$ and $9$ . Show that their product is equal to the fifth power of the single $G . M .$ between the given numbers.