+0

# Sum

0
6
2
+118

Compute the sum
$$1 + \frac{1}{5} + \frac{2}{5} + \frac{2}{25} + \frac{4}{25} + \frac{4}{125} + \frac{8}{125} + \frac{8}{625} + \dots$$

Apr 7, 2024

#1
+302
+3

This sum if really a combination of 2 geometric series.

First geometric series: $$1\over{5}$$,$$2\over{25}$$,$$4\over{125}$$, and so on

Second geometric series: $$2\over{5}$$, $$4\over{25}$$, $$8\over{125}$$, and so on

use: $$a_1\over1-r$$ formula: The sum of the first geometric series would be: $$1\over{5}$$$$\div$$$$3\over5$$=1/3

2nd would be: 2/5 $$\div$$ 3/5 = 2/3

1/3+2/3 = 1

We still didn't add the first term: 1

So 1+1 = 2

#1
+302
+3

This sum if really a combination of 2 geometric series.

First geometric series: $$1\over{5}$$,$$2\over{25}$$,$$4\over{125}$$, and so on

Second geometric series: $$2\over{5}$$, $$4\over{25}$$, $$8\over{125}$$, and so on

use: $$a_1\over1-r$$ formula: The sum of the first geometric series would be: $$1\over{5}$$$$\div$$$$3\over5$$=1/3

2nd would be: 2/5 $$\div$$ 3/5 = 2/3

1/3+2/3 = 1

We still didn't add the first term: 1

So 1+1 = 2

#2
+129830
0