Consider the sequences given in the table below. Find the least number, [tex]$n$[/tex], such that the [tex]$n$[/tex]th term of the geometric sequence is greater than the corresponding term in the arithmetic sequence.

\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
Term Number & 1 & 2 & 3 & 4 & 5 & 6 & [tex]$\ldots$[/tex] & [tex]$n$[/tex] \\
\hline
Arithmetic & 200 & 700 & 1200 & 1700 & 2200 & 2700 & [tex]$\ldots$[/tex] & \\
\hline
Geometric & 2 & 4 & 8 & 16 & 32 & 64 & [tex]$\ldots$[/tex] & \\
\hline
\end{tabular}

The geometric sequence is larger than the arithmetic sequence at the [tex]$\square$[/tex]th term.



Answer :

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.