If n geometric means are inserted between a a | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) If $n$ geometric means $g_1, g_2, \dots, g_n$ are inserted between two positive real numbers $a$ and $b$,then the sequence $a, g_1, g_2, \dots, g_

Vedclass · Accepted Answer

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

Vedclass · Answer

(C) If $n$ geometric means $g_1, g_2, \dots, g_n$ are inserted between two positive real numbers $a$ and $b$,then the sequence $a, g_1, g_2, \dots, g_n, b$ forms a Geometric Progression $(G.P.)$.Let $r$ be the common ratio of this $G.P.$ The total number of terms in this sequence is $n+2$.Thus,the last term $b$ can be expressed as $b = a \cdot r^{(n+2)-1} = a \cdot r^{n+1}$.Solving for $r$,we get $r = \left( \frac{b}{a} \right)^{\frac{1}{n+1}}$.The $n^{th}$ geometric mean is $g_n = a \cdot r^n$.Substituting the value of $r$,we get $g_n = a \left( \left( \frac{b}{a} \right)^{\frac{1}{n+1}} \right)^n = a \left( \frac{b}{a} \right)^{\frac{n}{n+1}}$.

If $n$ geometric means are inserted between $a$ and $b$,then the $n^{th}$ geometric mean will be:

Similar Questions

If the sum of an infinite $G.P.$ and the sum of the squares of its terms is $3$,then the common ratio of the first series is

The sum of the first $20$ terms of the sequence $0.7, 0.77, 0.777, \dots$ is

Determine the $25^{th}$ term of an $A.P.$ whose $9^{th}$ term is $-6$ and the common difference is $\frac{5}{4}$.

For a series $S = 1 - 2 + 3 - 4 + \dots$ up to $n$ terms,
Statement $-1$: The sum of the series is always dependent on the value of $n$,i.e.,whether it is even or odd.
Statement $-2$: The sum of the series is $-\frac{n}{2}$ when $n$ is any even integer.

If $\log_3 2, \log_3(2^x - 5)$ and $\log_3(2^x - 7/2)$ are in $A.P.$,then $x$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module