Respuesta :
Answer:
[tex]a_1=23/100\\\\\\r=1/100[/tex]
Explanation:
The question is incomplete and contains typing mistakes.
The correct question is:
The formula for the sum of an infinite geometric series,
[tex]S=\dfrac{a_1}{1-r}[/tex],
may be used to convert [tex]0.\overline{23}[/tex] (the the small line over 23 is used to indicate periodicity) to a fraction.
What are the values of [tex]a_1[/tex] and [tex]r[/tex] ?
Solution
Note that the number is equivalent to:
[tex]\dfrac{23}{100}+\dfrac{23}{10000}+\dfrac{23}{1000000}+ . . .[/tex]
Where the first term is 23/100 and the following is obtained multiplying the previous term by 1/100.
Thus, that is a geometric series with [tex]a_1=23\text { and }r=1/100[/tex]
Now, use the formula to confirm:
[tex]S=\dfrac{a_1}{1-r}\\\\\\S=\dfrac{23/100}{1-(1/100)}\\\\\\S=\dfrac{23}{100-1}\\\\\\S=\dfrac{23}{99}[/tex]
Use a calculator to verify that 23/99 = 0.232323 . . .
The value of a₁ is 0.23 and the value of r is 1/100 of infinite geometric series.
What is geometric series?
The geometric series defined as a series represents the sum of the terms in a finite or infinite geometric sequence. The successive terms in this series share a common ratio.
The nth term of a geometric progression is expressed as
Tₙ = arⁿ⁻¹
Where a is the first term, r is the common ratio
Given that,
The formula for the sum of an infinite geometric series as
S = a₁/(1 - r)
And a fraction 23 is used to represent repeated.
To determine the values of and r :
The given repeated fraction number is equivalent to:
0.2323232323... = 0.23 + 0.0023 + 0.000023 +...
0.2323232323... = 23/100 + 23/10000 + 23/1000000
The above series is in geometric progression.
Where the first term is 23/100
So, the value of a₁ is 0.23.
The value of the common ratio of geometric series can be calculated as:
r = 0.0023/0.23
r = 1/100
Hence, the value of a₁ is 0.23 and the value of r is 1/100.
Learn more about geometric series here:
brainly.com/question/21087466
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