Answer :
To find the number of terms in a geometric series given its sum, first term, and common ratio, we can use the formula for the sum of the first [tex]n[/tex] terms of a geometric series:
[tex]S_n = a \frac{1 - r^n}{1 - r}[/tex]
Where:
- [tex]S_n[/tex] is the sum of the series (46.5 in this case).
- [tex]a[/tex] is the first term of the series (24 here).
- [tex]r[/tex] is the common ratio (0.5).
- [tex]n[/tex] is the number of terms, which we need to find.
Given: [tex]S_n = 46.5[/tex], [tex]a = 24[/tex], and [tex]r = 0.5[/tex].
Plug these values into the formula:
[tex]46.5 = 24 \frac{1 - 0.5^n}{1 - 0.5}[/tex]
Simplify the denominator:
[tex]46.5 = 24 \frac{1 - 0.5^n}{0.5}[/tex]
[tex]46.5 = 48(1 - 0.5^n)[/tex]
Divide both sides by 48:
[tex]\frac{46.5}{48} = 1 - 0.5^n[/tex]
[tex]0.96875 = 1 - 0.5^n[/tex]
Rearrange to solve for [tex]0.5^n[/tex]:
[tex]0.5^n = 1 - 0.96875[/tex]
[tex]0.5^n = 0.03125[/tex]
To find [tex]n[/tex], take the logarithm of both sides:
[tex]\log(0.5^n) = \log(0.03125)[/tex]
Using the properties of logarithms, [tex]n \log(0.5) = \log(0.03125)[/tex].
Solve for [tex]n[/tex]:
[tex]n = \frac{\log(0.03125)}{\log(0.5)}[/tex]
Calculate the values:
[tex]n \approx \frac{-1.50515}{-0.30103}[/tex]
[tex]n \approx 5[/tex]
Therefore, there are 5 terms in the geometric series.