Answer to Question #217544 in Algebra for Ahmed Khan

Question

(b) For the given sequence 1/n, 1/2n, 1/4n,… Find the 16th term. Find sum of first sixteen terms where n is your arid number i.e. 19-arid-234 take n=234

Answer 1

(b)

$$\dfrac{1}{n},\dfrac{1}{2n},\dfrac{1}{4n},\dfrac{1}{8n}....$$

This is GP

So,

First term is $a=\dfrac{1}{n}$

and common ratio $r=\dfrac{1}{2}$

$p^{th}$ term of GP

$a_p=ar^{p-1}$

So, 16th term

$a_{16}=ar^{16-1}=ar^{15}=\dfrac{1}{n}\cdot (\dfrac{1}{2})^{15}=\dfrac{1}{32768n}$

and for n= 234

$a_{16}=\dfrac{1}{32768\times 234}=\dfrac{1}{7667712}$

and sum of p terms of GP

$S_p=\dfrac{a(1-r^p)}{1-r}$

So, sum of 16 terms of GP is

$S_{16}=\dfrac{\frac{1}{n}(1-(\frac{1}{2})^{16})}{1-\frac{1}{2}}=\dfrac{2}{n}(1-(\frac{1}{2})^{16})$

for n= 234

$S_{16}=\dfrac{2}{234}[1-\dfrac{1}{65536}]=\dfrac{21845}{2555904}$