To determine the number of terms in the series [tex]\(\frac{1}{3}+\frac{4}{3}+\frac{16}{3}+\ldots\)[/tex] that sum to 29127, we need to recognize that this is a geometric series. Let's solve this step-by-step:
1. Identify the first term ([tex]\(a\)[/tex]) and the common ratio ([tex]\(r\)[/tex]):
- The first term is [tex]\(a = \frac{1}{3}\)[/tex].
- The common ratio is [tex]\(r = 4\)[/tex], because each term is multiplied by 4 to get the next term.
2. Write the sum of the first [tex]\(n\)[/tex] terms of a geometric series:
The sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of a geometric series is given by:
[tex]\[
S_n = a \frac{1-r^n}{1-r}
\][/tex]
3. Plug in the known values:
- [tex]\(a = \frac{1}{3}\)[/tex]
- [tex]\(r = 4\)[/tex]
- [tex]\(S_n = 29127\)[/tex]
The formula becomes:
[tex]\[
29127 = \frac{1}{3} \frac{1-4^n}{1-4}
\][/tex]
4. Simplify the formula:
[tex]\[
29127 = \frac{1}{3} \frac{1-4^n}{-3}
\][/tex]
[tex]\[
29127 = \frac{1-4^n}{-9}
\][/tex]
5. Multiply both sides by [tex]\(-9\)[/tex] to eliminate the fraction:
[tex]\[
29127 \times -9 = 1 - 4^n
\][/tex]
[tex]\[
-262143 = 1 - 4^n
\][/tex]
6. Isolate [tex]\(4^n\)[/tex]:
[tex]\[
4^n = 1 + 262143
\][/tex]
[tex]\[
4^n = 262144
\][/tex]
7. Express 262144 as a power of 4:
We know that:
[tex]\[
262144 = 4^9
\][/tex]
because [tex]\(4^9 = (2^2)^9 = 2^{18}\)[/tex] and [tex]\(2^{18} = 262144\)[/tex].
8. So, [tex]\(n = 9\)[/tex]:
[tex]\[
4^n = 262144 \implies n = 9
\][/tex]
Thus, the number of terms in the series required to achieve a sum of 29127 is [tex]\(n = 9\)[/tex].
