Question

How do you find the sum of the infinite geometric series \[4,16,64,256,1024......\]

Answer 1

Hint: Sequence is basically a set of things that are in any order. Geometric sequence is a sequence where the ratio between each successive pair of terms is the same and it is abbreviated as GP. Next term of any sequence can be obtained by multiplying a constant number to the term before it. That constant number which is multiplied is known as the common ratio ( $ r $ ). Since all the geometric sequences follow the same pattern, we can write the same rule for finding the nth term for the sequence and for finding sum of terms of series.

Complete step by step solution:
 Given, \[4,16,64,256,1024......\]
We are given,
 $
  a = 4 \\
  r = 4 \\
  $
We’ll put these values in the formula of finding sum of infinite geometric series
 $ {S_\infty } = \dfrac{a}{{1 - r}} $
 $ \Rightarrow {S_\infty } = \dfrac{4}{{1 - 4}} $
 $ \Rightarrow {S_\infty } = \dfrac{4}{{ - 3}} $
 $ \Rightarrow {S_\infty } = - 1.33 $
So, the correct answer is “- 1.33 ”.

Note: Common ratio of any sequence can be obtained by dividing any of the two terms i.e. dividing the latter term from prior.
\[r = \dfrac{{{a_n}}}{{{a_{n - 1}}}}\]
 $ a $ represents first term of the series
 $ r $ represents common ratio
Also, Geometric Sequence can be both finite and infinite. The behavior of GP depends on the nature of common ratio. When a series has infinite series it is called infinite series whereas when it has finite terms it is called infinite series.
If the common ratio is a positive number, the sequence will progress towards infinity.
If the common ratio is a negative number, the sequence will regress towards negative infinity.
Sum of terms of series can be calculated when the common ratio in the infinite series lies between $ - 1\;and\;1 $ .