Question

There are three geometrical means between a and b. If the first and the third means are 25 and 125 respectively, find a and b.​

Answer 1

Answer:

It is very simple just you have to focus on the question

Step-by-step explanation:

1st of all just draw a basic concept about mean...It to boring to write hope that you will understand from photo

Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

There are 3 geometric means between a and b.

So, Let assume that,

$\rm :\longmapsto\:3 \: geometric \: means \: be \: g_1, \: g_2 \: and \: g_3$

So, that,

$\rm :\longmapsto\:a, \: g_1, \: g_2, \: g_3, \: b \: are \: in \: GP$

Further, given that

$\rm :\longmapsto\:\boxed{ \tt{ \: g_1 = 25 \: }}$

and

$\rm :\longmapsto\:\boxed{ \tt{ \: g_3 = 125 \: }}$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an geometric sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\: {r}^{n \: - \: 1} }}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

aₙ is the nᵗʰ term.

a is the first term of the sequence.

n is the no. of terms.

r is the common ratio.

Tʜᴜs,

$\rm :\longmapsto\:g_1 = 25$

$\rm \implies\: {ar}^{2 - 1} = 25$

$\rm \implies\: {ar}^{} = 25 - - - - (1)$

Also,

$\rm :\longmapsto\:g_3 = 125$

$\rm \implies\: {ar}^{4 - 1} = 125$

$\rm \implies\: {ar}^{3} = 125$

$\rm :\longmapsto\:ar \times {r}^{2} = 125$

$\rm :\longmapsto\:25 \times {r}^{2} = 125$

$\rm :\longmapsto\:{r}^{2} = 5$

$\bf\implies \:r \: = \: \pm \: \sqrt{5}$

So, on substituting the value of r in equation (1), we get

$\red{\rm :\longmapsto\:When \: r \: = \: \sqrt{5} }$

So,

$\rm \implies\: a \times \sqrt{5} = 25$

$\rm \implies\: a = 5 \sqrt{5}$

$\red{\rm :\longmapsto\:When \: r \: = \: - \: \sqrt{5} }$

So,

$\rm \implies\: - a \times \sqrt{5} = 25$

$\rm \implies\: a \: = \: - \: 5 \sqrt{5}$

Now,

$\rm :\longmapsto\:b$

$\rm \:  =  \:a_5$

$\rm \:  =  \: {ar}^{4}$

So, Case - 1

$\red{\begin{gathered}\begin{gathered}\bf\:\rm :\longmapsto\:\begin{cases} &\sf{a = 5 \sqrt{5} } \\ \\ &\sf{r = \sqrt{5} } \end{cases}\end{gathered}\end{gathered}}$

Thus,

$\rm :\longmapsto\:b = 5 \sqrt{5} \times {( \sqrt{5} )}^{4} = 5 \sqrt{5} \times 25 = 125 \sqrt{5}$

Case - 2

$\red{\begin{gathered}\begin{gathered}\bf\:\rm :\longmapsto\:\begin{cases} &\sf{a \: = \: - \: 5 \sqrt{5} } \\ \\ &\sf{r \: = \: - \: \sqrt{5} } \end{cases}\end{gathered}\end{gathered}}$

Thus,

$\rm :\longmapsto\:b = - 5 \sqrt{5} \times {( - \sqrt{5} )}^{4} = - 5 \sqrt{5} \times 25 = - 125 \sqrt{5}$