Question

Evaluate the following geometric sum.

1/2 + 1/10 + ( 1/50) + (1/250 ) + midline ellipsis + (1/31,250)

Answer 1

To evaluate the geometric sum 1/2 + 1/10 + (1/50) + (1/250) + midline ellipsis + (1/31,250), we can use the formula for the sum of an infinite geometric series. The sum is 5/8.

Step-by-step explanation:

To evaluate the given geometric sum 1/2 + 1/10 + (1/50) + (1/250) + (ellipsis) + (1/31,250), we need to find the sum of an infinite geometric series. This series has a common ratio of 1/5.

Here's how you can find the sum:

1. Start with the first term, which is 1/2.
2. Use the formula for the sum of an infinite geometric series:
   Sum = a / (1 - r), where a is the first term and r is the common ratio.
3. Plug in the values:
   Sum = (1/2) / (1 - 1/5)
4. Simplify the expression:
   Sum = (1/2) / (4/5) = (5/2) / 4 = 5/8

Therefore, the sum of the given geometric series is 5/8.

Answer 2

39062/62,500

Step-by-step explanation:

Given the following geometric progression; 1/2 + 1/10 + ( 1/50) + (1/250 ) + ... + (1/31,250),the sum of the arithmetic geometric progression is expressed using the formula below;

Sn = a(1-rⁿ)/1-r for r less than 1

r is the common ratio

n is the number of terms

a is the first term of the series

In between the mid-line ellipsis there are 2 more terms, making the total number of terms n to be 7]

common ratio = (1/10)/(1/2) = (1/50)/(1/10) = (1/250)/(1/50) = 1/5

a = 1/2

Substituting the given values into the equation above

S7 = 1/2{1 - (1/5)⁷}/1 - 1/5

S7 = 1/2(1- 1/78125)/(4/5)

S7 = 1/2 (78124/78125)/(4/5)

S7 = 78124/156,250 * 5/4

S7 = 390,620/625000

S7 = 39062/62,500

Hence the geometric sum is 39062/62,500