Question

How do you find the sum of the geometric series $2 - 10 + 50 - ...$ to $6$ terms?

Answer 1

Hint: The given type of series given in the question is a geometric series or the terms are in Geometric Progression. In a geometric progression, each term is multiplied by a common ratio to get the next term. The terms of such series is given by, $a,{\kern 1pt} {\kern 1pt} {\kern 1pt} ar,{\kern 1pt} {\kern 1pt} {\kern 1pt} a{r^2},...$ , where $r$ is the common ratio. We can use the formula to find the sum of the series up to $n$ terms given by, ${S_n} = \dfrac{{a({r^n} - 1)}}{{(r - 1)}}$.

Formula used:
${S_n} = \dfrac{{a({r^n} - 1)}}{{(r - 1)}}$

Complete step by step solution:
We are given a series $2 - 10 + 50 - ...$
We are given that this series is a Geometric Progression.
We can find the common ratio of the series by dividing the consecutive terms.
We can calculate the common ratio as $r = \dfrac{{ - 10}}{2} = \dfrac{{50}}{{ - 10}} = - 5$
Thus, the given series is a Geometric Progression (GP) with common ratio $r = - 5$
Now we can use the formula to find the sum of the series up to $n$ terms given by, ${S_n} = \dfrac{{a({r^n} - 1)}}{{(r - 1)}}$.
We have to find the sum of the series up to $6$ terms. So, $n = 6$.
Also, the first term of the series is $a = 2$.
Putting all the values in the above formula, we get:
\[
  {S_n} = \dfrac{{a({r^n} - 1)}}{{(r - 1)}} \\
   \Rightarrow {S_n} = \dfrac{{2\{ {{( - 5)}^6} - 1\} }}{{( - 5 - 1)}} \\
   \Rightarrow {S_n} = \dfrac{{2\{ 15625 - 1\} }}{{( - 6)}} \\
   \Rightarrow {S_n} = \dfrac{{2 \times (15624)}}{{ - 6}} \\
   \Rightarrow {S_n} = \dfrac{{15624}}{{ - 3}} \\
   \Rightarrow {S_n} = - 5208 \\
\]
Thus, the sum of the given infinite series is $ - 5208$.

Note: We can use the formula to find the sum of the series up to $n$ terms. We can calculate the sum of the series without knowing all the terms, we only need at most three terms to calculate the common ratio. Though we had to find the sum, the answer can be negative when the common ratio is a negative number or all the terms are negative.