Question

How do you find the sum of the finite geometric sequence of $ \sum {8{{\left( { - \dfrac{1}{4}} \right)}^{i - 1}}} $ from $ i = 0$ to $10? $

Answer 1

Hint: As we know that a geometric sequence is a sequence of nonzero numbers where each term after the first is found by multiplying the previous one by a fixed and nonzero number. And when we know the number of terms in a geometric sequence then it is called a finite geometric sequence. The general formula of finite geometric sequence is $ \sum\limits_{i = 0}^n {a{r^n}} = \dfrac{{a(1 - {r^n})}}{{1 - r}} $ .

Complete step by step solution:
We will solve the above given question by applying the general formula of the fine geometric sequence. First we will write the question in the form of formula i.e.
$ \sum\limits_{i = 0}^{10} {8{{\left( { - \dfrac{1}{4}} \right)}^{i - 1}} = } 8\sum\limits_{i = 0}^{10} {{{\left( { - \dfrac{1}{4}} \right)}^{i - 1}}} $ .
We can break down the powers with the same base i.e.
$ 8\sum\limits_{i = 0}^{10} {{{\left( { - \dfrac{1}{4}} \right)}^{ - 1}}{{\left( { - \dfrac{1}{4}} \right)}^i}} $ .
This can be written as .
$ 8{\left( { - \dfrac{1}{4}} \right)^{ - 1}}\sum\limits_{i = 0}^{10} {{{\left( { - \dfrac{1}{4}} \right)}^i}} $ .
By applying the general formula above to the latter sum we note that we can let
$ n = 10,a = 1 $ and $ r = \left( { - \dfrac{1}{4}} \right) $ .
We substitute this value in the formula and we have
$ 8{\left( { - \dfrac{1}{4}} \right)^{ - 1}} \times \dfrac{{1\left\{ {1 - {{\left( { - \dfrac{1}{4}} \right)}^{10}}} \right\}}}{{1 - \left( { - \dfrac{1}{4}} \right)}} $ .
We will now solve it by multiplying the above expression with the diffraction of
$ \dfrac{{{{( - 4)}^{10}}}}{{{{( - 4)}^{10}}}} $ .
By applying the power of exponent we can write
$ 8{\left( {\dfrac{{ - 1}}{4}} \right)^{ - 1}} = 8\left( { - 4} \right) $ , since negative power of inverse reverse the dfarction.
It can be written as
$ 8( - 4)\dfrac{{1 - {{\left( { - \dfrac{1}{4}} \right)}^{10}}}}{{1 - \left( { - \dfrac{1}{4}} \right)}} \cdot \dfrac{{{{( - 4)}^{10}}}}{{{{( - 4)}^{10}}}} $ .
On further solving
$ - 32 \cdot \dfrac{{{{( - 4)}^{10}} - {{(1)}^{10}}}}{{{{( - 4)}^{10}} - {{( - 4)}^{9}}}} \Rightarrow - 32\dfrac{{{4^{10}} - 1}}{{{4^{10}} + {4^9}}} $ .
By breaking down the powers we have $ - \dfrac{{1048576 - 1}}{{5 \times 8192}} $ .
Hence the required value is $ \sum {8{{\left( { - \dfrac{1}{4}} \right)}^{i - 1}}} = - \dfrac{{209715}}{{8192}} $ .
So, the correct answer is “ $ - \dfrac{{209715}}{{8192}} $ .”.

Note: Before solving this type of question we should have the proper knowledge of geometric sequence or progression and their formulas. In the above question the basic rule of exponential form is also used. We should solve this question carefully by applying the rule of exponents as it can make the solution easier.