Express the repeating decimal 0.2323 as a geometric series, and write its sum as the ratio of two
integers
a = .23
r = .01
S = .23/(1-.01) = .23/.99 = 23/99
23/99
To express the repeating decimal 0.2323 as a geometric series, we can first observe the pattern in the decimal. If we separate the repeating part, we can see that it consists of the digits 23 repeating. Let's call this repeating part "r." So, we have:
0.2323 = 0.23 + 0.0003 + 0.000003 + ...
Now, we can rewrite this in a different form by breaking up the repeating decimal into smaller parts:
0.2323 = 0.23 + 0.0023 + 0.000023 + ...
Next, notice that each part of the series can be obtained by multiplying the previous part by a common ratio. In this case, the common ratio, denoted by "r," is 0.01. So, we have:
0.2323 = 0.23(1 + 0.01 + 0.0001 + ...)
Now, we have an expression that represents 0.2323 as a geometric series:
0.2323 = 0.23(1 + r + r^2 + ...)
To find the sum of this geometric series, we use the formula for the sum of an infinite geometric series:
S = a / (1 - r)
Where "S" is the sum of the series, "a" is the first term (0.23), and "r" is the common ratio (0.01).
Substituting the values into the formula, we get:
S = 0.23 / (1 - 0.01)
Now, we can simplify this expression by performing the calculations:
S = 0.23 / 0.99
To write the sum as the ratio of two integers, we can multiply both the numerator and denominator by 100 to eliminate the decimal:
S = (0.23 * 100) / (0.99 * 100)
Simplify this expression further:
S = 23 / 99
Therefore, the sum of the geometric series 0.2323 is 23/99, which can be expressed as a ratio of two integers.