Sure! To find the number of terms in a geometric series, we need to use the formula for the sum 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,
- [tex]\( a \)[/tex] is the first term,
- [tex]\( r \)[/tex] is the common ratio,
- [tex]\( n \)[/tex] is the number of terms.
Given:
- The sum of the series [tex]\( S_n = 31.5 \)[/tex],
- The first term [tex]\( a = 16 \)[/tex],
- The common ratio [tex]\( r = 0.5 \)[/tex].
To find the number of terms [tex]\( n \)[/tex], we rearrange the formula to solve for [tex]\( n \)[/tex]:
1. Multiply both sides of the equation by [tex]\( 1 - r \)[/tex]:
[tex]\[ S_n (1 - r) = a (1 - r^n) \][/tex]
2. Substitute the given values:
[tex]\[ 31.5 (1 - 0.5) = 16 (1 - 0.5^n) \][/tex]
3. Calculate [tex]\( 1 - 0.5 \)[/tex]:
[tex]\[ 31.5 \times 0.5 = 16 \times (1 - 0.5^n) \][/tex]
[tex]\[ 15.75 = 16 \times (1 - 0.5^n) \][/tex]
4. Divide both sides by 16:
[tex]\[ \frac{15.75}{16} = 1 - 0.5^n \][/tex]
[tex]\[ 0.984375 = 1 - 0.5^n \][/tex]
5. Subtract 1 from both sides and multiply by -1:
[tex]\[ 0.5^n = 1 - 0.984375 \][/tex]
[tex]\[ 0.5^n = 0.015625 \][/tex]
6. Now, take the logarithm of both sides (base 10 or natural logarithm):
[tex]\[ \log(0.5^n) = \log(0.015625) \][/tex]
[tex]\[ n \log(0.5) = \log(0.015625) \][/tex]
7. Solve for [tex]\( n \)[/tex] by dividing both sides by [tex]\( \log(0.5) \)[/tex]:
[tex]\[ n = \frac{\log(0.015625)}{\log(0.5)} \][/tex]
8. Calculate the logarithms and solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{\log(0.015625)}{\log(0.5)} = 6 \][/tex]
So, there are 6 terms in the series.
