The first three terms of a geometric progress | vedclass

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

Question

(B) Let the three terms be $\frac{b}{r}, b, br$ where $r$ is the common ratio.Given that the harmonic mean of $a$ and $b$ is $12$:$\frac{2ab}{a+b} = 1

Vedclass · Accepted Answer

$8$

Vedclass · Answer

(B) Let the three terms be $\frac{b}{r}, b, br$ where $r$ is the common ratio.Given that the harmonic mean of $a$ and $b$ is $12$:$\frac{2ab}{a+b} = 12 \implies \frac{2(\frac{b}{r})b}{\frac{b}{r}+b} = 12 \implies \frac{2b}{1+r} = 12 \implies b = 6(1+r) \dots (1)$Given that the harmonic mean of $b$ and $c$ is $36$:$\frac{2bc}{b+c} = 36 \implies \frac{2b(br)}{b+br} = 36 \implies \frac{2br}{1+r} = 36 \implies br = 18(1+r) \dots (2)$Dividing $(2)$ by $(1)$:$\frac{br}{b} = \frac{18(1+r)}{6(1+r)} \implies r = 3$Substituting $r=3$ into $(1)$:$b = 6(1+3) = 24$Since $a = \frac{b}{r} = \frac{24}{3} = 8$.

The first three terms of a geometric progression are $a, b, c$. If the harmonic mean of $a$ and $b$ is $12$ and the harmonic mean of $b$ and $c$ is $36$,then $a = \dots$

Similar Questions

Let $a_{n}$ be the $n^{\text{th}}$ term of a $G$.$P$. of positive terms. If $\sum_{n=1}^{100} a_{2n+1} = 200$ and $\sum_{n=1}^{100} a_{2n} = 100$,then $\sum_{n=1}^{200} a_{n}$ is equal to:

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of a geometric progression are $a, b, c$ respectively,then $a^{q-r} \cdot b^{r-p} \cdot c^{p-q} = \dots\dots$

The sum of the series $5.05 + 1.212 + 0.29088 + \dots \infty$ is

If $S_n$ is the sum of $n$ terms of a geometric progression with first term $a$ and common ratio $r$,what is the sum of $S_1 + S_3 + S_5 + \dots + S_{2n-1}$?

If $G$ is the geometric mean of $x$ and $y$,then $\frac{1}{G^2 - x^2} + \frac{1}{G^2 - y^2} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module