Question

If the first term of a sequence is $ a = 2 $ and the common ratio between the terms are $ r = - \dfrac{2}{3} $ . Then find the sum of the first $ 6 $ terms of the geometric progression.

Answer 1

Hint: Geometric progression is similar to arithmetic progression, it has a fixed pattern. The pattern here is in the common ratio that is ratio of the current term to its previous term, this value will be the same between every pair of consecutive numbers in the sequence. Since the first term of the sequence and common ratio are given, find the sum using the formula for the sum of $ n $ terms of a geometric progression, here $ n = 6 $ .

Complete step-by-step answer:
Let us first write down the observations from the question itself.
The data given to us says that;
 $ a = 2 $ , here $ a $ denotes the first term of the required sequence.
 $ r = - \dfrac{2}{3} $ , it is common ratio or ratio between current term and its previous term
To find: $ S_6 $ that is the sum of first $ 6 $ terms in the geometric sequence.
To find the sum $ S_6 $ we can use the formula for $ n $ terms in a geometric progression, here since $ r < 1 $ , we use the case of sum where $ r < 1 $ , that is;
 $ \Rightarrow Sn = \dfrac{{a.(1 - {r^n})}}{{(1 - r)}} $
Substituting the given values of $ a = 2 $ and $ r = - \dfrac{2}{3} $ we get;
 $ \Rightarrow S_6 = \dfrac{{2 \times [1 - {{( - \dfrac{2}{3})}^6}]}}{{[1 - ( - \dfrac{2}{3})]}} $
 $ \Rightarrow S_6 = \dfrac{{2 \times [1 - {{( - \dfrac{2}{3})}^6}]}}{{[\dfrac{5}{3}]}} $
 $ \Rightarrow S_6 = \dfrac{6}{5} \times (\dfrac{{729 - 64}}{{{3^6}}}) $
 $ \Rightarrow S_6 = \dfrac{6}{5} \times (\dfrac{{665}}{{729}}) $
 $ \Rightarrow S_6 = \dfrac{{266}}{{243}} $
Therefore the sum of the first $ 6 $ terms ( $ S_6 $ )of this geometric progression is $ \dfrac{{266}}{{243}} $ .
So, the correct answer is “ $ \dfrac{{266}}{{243}} $ ”.

Note: Similar to a geometric progression we have another sequence called the arithmetic progression. It is a sequence of numbers which have a common difference between every pair of consecutive integers. So if the first term and common difference of a sequence are given, then we can utilize them to find any term in the sequence. Each term can be denoted in general as $ an $ , here we can find $ an $ using a general formula that is;
 $ an = a + (n - 1)d $ , where $ a $ is the first term of the sequence, $ d $ is the common difference and $ n $ is the number of terms.