To solve the problem, we need to compare the [tex]$n$[/tex]th terms of the two given sequences and find the smallest [tex]$n$[/tex] such that the [tex]$n$[/tex]th term of the geometric sequence exceeds the [tex]$n$[/tex]th term of the arithmetic sequence.
1. Understand the sequences:
- The terms of the arithmetic sequence are given by:
[tex]\[
A_n = 200 + (n-1) \cdot 500
\][/tex]
- The terms of the geometric sequence are given by:
[tex]\[
G_n = 2 \cdot 2^{n-1}
\][/tex]
Simplifying the geometric sequence formula:
[tex]\[
G_n = 2^n
\][/tex]
2. Set the inequality:
We need to find the smallest [tex]$n$[/tex] such that:
[tex]\[
G_n > A_n
\][/tex]
Substituting the expressions for [tex]$A_n$[/tex] and [tex]$G_n$[/tex], we get:
[tex]\[
2^n > 200 + (n-1) \cdot 500
\][/tex]
3. Solve for [tex]\( n \)[/tex]:
Given in the solution, the sequences first satisfy the inequality when [tex]\( n = 13 \)[/tex]. At this stage:
- The [tex]$13$[/tex]th term of the arithmetic sequence (using the formula [tex]\( A_n = 200 + (n-1) \cdot 500 \)[/tex]):
[tex]\[
A_{13} = 200 + 12 \cdot 500 = 200 + 6000 = 6200
\][/tex]
- The [tex]$13$[/tex]th term of the geometric sequence (using the formula [tex]\( G_n = 2^n \)[/tex]):
[tex]\[
G_{13} = 2^{13} = 8192
\][/tex]
4. Comparison at the [tex]$13$[/tex]th term:
Clearly,
[tex]\[
8192 > 6200
\][/tex]
This shows the geometric sequence is larger than the arithmetic sequence starting from the [tex]$13$[/tex]th term.
Thus, the geometric sequence becomes larger at the [tex]\(13\)[/tex]th term.
