If α ,β ,gamma are geometric means between ca,ab;ab,bc;bc,ca respectively where a,b,c are in A.P.,

English

Gujarati

Hindi

8. Sequences and Series

medium

If $\alpha ,\;\beta ,\;\gamma $ are the geometric means between $ca,\;ab;\;ab,\;bc;\;bc,\;ca$ respectively where $a,\;b,\;c$ are in A.P., then ${\alpha ^2},\;{\beta ^2},\;{\gamma ^2}$ are in

A

$A.P.$

B

$H.P.$

C

$G.P.$

D

None of the above

Solution

(a) By hypothesis, ${\alpha ^2} = {a^2}bc,\;{\beta ^2} = {b^2}ca,\;{\gamma ^2} = {c^2}ab$ and $2b = a + c$.

.Hence ${2^{n – 1}} > 100$ are in A.P.

Standard 11

Mathematics

Share

Similar Questions

Write the first five terms of the sequences whose $n^{t h}$ term is $a_{n}=n \frac{n^{2}+5}{4}$

medium

The $A.M.$ of a $50$ set of numbers is $38$. If two numbers of the set, namely $55$ and $45$ are discarded, the $A.M.$ of the remaining set of numbers is

medium

If $(b+c),(c+a),(a+b)$ are in $H.P$ , then $a^2,b^2,c^2$ are in…….

normal

If ${S_k}$ denotes the sum of first $k$ terms of an arithmetic progression whose first term and common difference are $a$ and $d$ respectively, then ${S_{kn}}/{S_n}$ be independent of $n$ if

hard

The ratio of the sums of $m$ and $n$ terms of an $A.P.$ is $m^{2}: n^{2} .$ Show that the ratio of $m^{ th }$ and $n^{ th }$ term is $(2 m-1):(2 n-1)$

medium