Question

In a GP, first term is a and the common ratio is r. If A and H are the arithmetic means and harmonic means respectively for the first n terms of GP. Them A * H is equal to

• A.
• B. a
• C.
• D. None of these

Answer 1

The correct option is A

Let the GP be a,ar,a$r^2$,.............a$r^{n-1}$

Arithmetic means of these number A = $\frac{a+ar+a{r}^2+.............+a{r}^{n-1}}{n}$ = $\frac{a(1-r^n)}{n(1-r)}$

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

= $\frac{n}{\frac{1}{a}(1+(\frac{1}{r})+(\frac{1}{r}{)}^2+.....\left(\frac{1}{r}{)}^{n-1}\right)}$

$=\frac{na}{\frac{1(1-{\left(\frac{1}{r}\right)}^{n-1})}{a(1-\frac{1}{r})}}$

= $\frac{na}{\frac{(r^n-1)}{r^n}\times \frac{1}{\frac{(r-1)}{r}}}$

= $\frac{na\times \left(r^{n-1}\right)(r-1)}{(r^n-1)}$

AH = $\frac{a}{n}\frac{(1-r^n)}{n(1-r)}timesna\times \frac{r^{n-1}\times (r-1)}{(r^n-1)}$

= $a^2\times r^{n-1}$ --------(A)