Question

How do you find $ {S_{11}} $ for the geometric series $ 1 + 3 + 9 + 27 + ...? $

Answer 1

Hint: As we know that a series is defined as an expression in which infinitely many terms are added one after the another to a given starting quantity. In this question we are given a GP series or Geometric progression series. It is a sequence of non zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. We will first find the common ratio of the given series and then solve it on the basis of that.

Complete step by step solution:
In the given question we have $ 1 + 3 + 9 + 27 + ... $ .
We know that the general form of the Geometric series is $ {a_1} + {a_2}r + {a_3}{r^2} + ...a{r^n} $ , where $ {a_1} $ is the first term, $ {a_2} $ is the second term and so on… and $ r $ is the common ratio.
So in the given series we have $ {a_1} = 1 $ and $ {a_2} = 3 $ .
We can find the common ratio by the formula $ r = \dfrac{{{a_2}}}{{{a_1}}} $ .
So it gives us $ r = \dfrac{3}{1} = 3 $ , and we have $ n = 11 $ . Here we have $ r > 1 $ .
We know that the sum of geometric series is $ {S_{}} = \dfrac{{a({r^n} - 1)}}{{r - 1}} $ , where $ n = 11 $ .
Now by putting the values we have
 $ 1.\dfrac{{1 - {3^{11}}}}{{1 - 3}} \Rightarrow \dfrac{{1 - 177147}}{{ - 2}} $ .
On further solving we have
 $ {S_{11}} = - \dfrac{{177146}}{{ - 2}} = 88573 $ .
Hence the sum of the given geometric series i.e. $ {S_{11}} = 88573 $ .
So, the correct answer is “88573”.

Note: We should note that if the geometric series is finite then we take the formulas as $ {S_n} = \dfrac{{a({r^n} - 1)}}{{r - 1}};\,r > 1 $ and if $ r < 1 $ , then the formula is $ {S_n} = \dfrac{{a(1 - {r^n})}}{{1 - r}} $ . Before solving such questions we should be well aware of the geometric progressions and their formulas. We should do the calculations very carefully especially while finding the sums of terms using the formula. It should be noted that the sum of n terms of arithmetic progression is given by $ \dfrac{1}{2}\left( {2a + (n - 1)d} \right) $ .