How do you find the sum of the finite geometric sequence of #Sigma 10(1/5)^n# from n=0 to 20?

Question

Emily Ammerman · Accepted Answer

#12.45# (2 .d.p)

First calculate the first three terms: #10(1/5)^0=color(blue)(10) , 10(1/5)^1= color(blue)(2) , 10(1/5)^2= color(blue)(2/5)# Find the common ratio: #2/10 = (2/5)/2=1/5# The sum of a geometric sequence is: #a((1-r^n)/(1-r))# Where a is the first term, n is the nth term and r is the common ratio. So: #10((1-(1/5)^20)/(1-(1/5)))=12.4999999999999737856#

Christopher Agresta · Answer

undefined To find the sum of the finite geometric sequence $ \sum_{n=0}^{20} 10\left(\frac{1}{5}\right)^n $, you can use the formula for the sum of a finite geometric series:

$$ S = \frac{a(1 - r^{n+1})}{1 - r} $$

where $ S $ is the sum of the series, $ a $ is the first term, $ r $ is the common ratio, and $ n $ is the number of terms.

In this sequence:
- The first term $ a $ is $ 10 $
- The common ratio $ r $ is $ \frac{1}{5} $
- The number of terms $ n $ is $ 20 $

Plug these values into the formula:

$$ S = \frac{10(1 - \left(\frac{1}{5}\right)^{20+1})}{1 - \frac{1}{5}} $$

$$ S = \frac{10(1 - \frac{1}{5}^{21})}{\frac{4}{5}} $$

$$ S = \frac{10(1 - \frac{1}{5^{21}})}{\frac{4}{5}} $$

$$ S = \frac{10(1 - \frac{1}{9765625})}{\frac{4}{5}} $$

$$ S = \frac{10 \times 9765624}{4} $$

$$ S = 2441406 $$

Therefore, the sum of the finite geometric sequence is $ 2441406 $.