Question

Let there be a GP whose first term is $a$ and the common ratio is $r$. If A and H are the arithmetic mean and the harmonic mean respectively for the first $n$ terms of the GP, $A.H$ is equal to

• A. $a^2{r}^{n-1}$
• B. $a{r}^n$
• C. $a^2{r}^n$
• D. none of these

Answer 1

Answer: (A)

$a^2{r}^{n-1}$

Given

$A=\frac{a+ar+a{r}^2+..+a{r}^{n-1}}{n}$

$=\frac{a}{n}\frac{r^n-1}{r-1}$
and

$\frac{n}{H}=\frac{1}{a}+\frac{1}{ar}+\frac{1}{a{r}^2}+..+\frac{1}{a{r}^{n-1}}$
$=\frac{1}{a}\left(\frac{1-\frac{1}{r^n}}{1-\frac{1}{r}}\right)$
$=\frac{1}{a{r}^{n-1}}\left(\frac{r^n-1}{r-1}\right)$
$=\frac{1}{a{r}^{n-1}}\frac{nA}{a}$
$\Rightarrow \frac{n}{H}=\frac{1}{a{r}^{n-1}}\frac{nA}{a}$
$\Rightarrow HA=a^2{r}^{n-1}$

Hence, option A is correct.