How many terms of the G.P. 3 , ? 3 2 , ? 3 4 , ... are needed to give the sum 3069 512 ?

Question

How many terms of the G.P. $3, \frac{3}{2}, \frac{3}{4}$ , ... are needed to give the sum $\frac{3069}{512}$ ?

Smart Achievers · Accepted Answer

(4)

Let $n$ be the number of terms needed. Given that $a = 3, r = \frac{1}{2}$ and $S_{n} = \frac{3069}{512}$

Now, $S_{n} = \frac{a (1 - r^{n})}{1 - r}$ $\Rightarrow$ $\frac{3069}{512} = \frac{3 (1 - \frac{1}{2^{n}})}{1 - \frac{1}{2}} = 6 (1 - \frac{1}{2^{n}})$

$\Rightarrow \frac{3069}{3072} = 1 - \frac{1}{2^{n}}$ $\Rightarrow \frac{1}{2^{n}} = 1 - \frac{3069}{3072} = \frac{1}{1024}$

$\Rightarrow 2^{n} = 1024 = 2^{10}$

$\Rightarrow n = 10$

Solution

Q.
How many terms of the G.P. $3, \frac{3}{2}, \frac{3}{4}$ , ... are needed to give the sum $\frac{3069}{512}$

1 9

2 12

3 8

4 10

Want Us to Email you About Special Offers & Updates?

Solution

Q. How many terms of the G.P. 3, 32, 34, ... are needed to give the sum 3069512

1 9

2 12

3 8

4 10

Ans. (4) Let n be the number of terms needed. Given that a=3, r=12 and Sn=3069512 Now, Sn=a(1-rn)1-r ⇒ 3069512=3(1-12n)1-12=6(1-12n) ⇒30693072=1-12n⇒12n=1-30693072=11024 ⇒2n=1024=210 ⇒n=10

Want Us to Email you About Special Offers & Updates?

Q.
How many terms of the G.P. $3, \frac{3}{2}, \frac{3}{4}$ , ... are needed to give the sum $\frac{3069}{512}$