Question

1. which is larger, the 10th term of an arithmetic sequence that begins with the terms 0 and 100 or the 10th term of a geometric sequence that begins with the terms 5 and 10? show work that justifies your answer.

Explanation:

Step1: Find arithmetic sequence common difference

The arithmetic sequence starts with $a_1=0$, $a_2=100$. The common difference $d = a_2 - a_1 = 100 - 0 = 100$.

Step2: Calculate 10th term of arithmetic sequence

Use arithmetic sequence formula $a_n = a_1 + (n-1)d$. For $n=10$:
$a_{10} = 0 + (10-1)\times100 = 9\times100 = 900$

Step3: Find geometric sequence common ratio

The geometric sequence starts with $g_1=5$, $g_2=10$. The common ratio $r = \frac{g_2}{g_1} = \frac{10}{5} = 2$.

Step4: Calculate 10th term of geometric sequence

Use geometric sequence formula $g_n = g_1\times r^{n-1}$. For $n=10$:
$g_{10} = 5\times 2^{10-1} = 5\times 2^9 = 5\times512 = 2560$

Step5: Compare the two 10th terms

Compare $900$ and $2560$: $2560 > 900$.

Answer:

The 10th term of the geometric sequence (2560) is larger than the 10th term of the arithmetic sequence (900).