Consider the arithmetic sequence $$ 30,38,46,54 \ldots $$ and the geometric sequence $$ 2,3,4.5,6.75 $$. Which sequence has the larger value at position 13 ? What is the difference between the larger and smaller value, correct to two decimal places?

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We first find the 13th term of the arithmetic sequence and the geometric sequence.

**Arithmetic Sequence:**  
The sequence is given by  
$$
a_n = 30 + (n-1)\times 8.
$$
For $$ n = 13 $$:
$$
a_{13} = 30 + (13-1)\times 8 = 30 + 12\times 8 = 30 + 96 = 126.
$$

**Geometric Sequence:**  
The sequence is given by  
$$
a_n = 2 \times (1.5)^{\,n-1}.
$$
For $$ n = 13 $$:
$$
a_{13} = 2 \times (1.5)^{12}.
$$
We now compute $$(1.5)^{12}$$. Using a calculator or by successive multiplication, we find:
$$
(1.5)^{12} \approx 129.746337890625.
$$
Thus,
$$
a_{13} \approx 2 \times 129.746337890625 \approx 259.49267578125.
$$

**Comparison and Difference:**  
The arithmetic sequence has a 13th term of $$126$$ and the geometric sequence approximately $$259.49$$. Therefore, the geometric sequence has the larger 13th term.

The difference is:
$$
259.49267578125 - 126 \approx 133.49267578125.
$$
Rounded to two decimal places, the difference is approximately:
$$
133.49.
$$

$$
\textbf{Answer: The geometric sequence has the larger value, and the difference is approximately }133.49.
$$
The geometric sequence has a larger 13th term, and the difference between the two sequences at that position is approximately 133.49.

Consider the arithmetic sequence and the geometric sequence .
Which sequence has the larger value at position 13 ? What is the difference between the larger
and smaller value, correct to two decimal places?

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Consider the arithmetic sequence and the geometric sequence . Which sequence has the larger value at position 13 ? What is the difference between the larger and smaller value, correct to two decimal places?