To determine the 10th term of the sequence, let's analyze the given terms:

The sequence starts with:

• $a_1 = 40$
• $a_2 = 24$
• $a_3 = \frac{72}{5} = 14.4$

This sequence appears to follow a pattern where each term is multiplied by a common ratio $r$ to find the next term. Let’s verify if this is a geometric sequence by calculating the ratio between consecutive terms:

$r = \frac{a_2}{a_1} = \frac{24}{40} = 0.6$

To confirm, let’s see if the second ratio matches: $r = \frac{a_3}{a_2} = \frac{14.4}{24} = 0.6$

Since both ratios are equal, this is a geometric sequence with a common ratio $r = 0.6$.

In a geometric sequence, the $n$-th term is given by: $a_n = a_1 \cdot r^{(n-1)}$

Finding the 10th term:

Substitute $a_1 = 40$, $r = 0.6$, and $n = 10$: $a_{10} = 40 \cdot (0.6)^{9}$

Calculating $(0.6)^9$: $(0.6)^9 \approx 0.01008$

Now, multiply this by 40: $a_{10} = 40 \cdot 0.01008 = 0.4032$

Rounded to the nearest thousandth, the 10th term is: $\boxed{0.403}$

Would you like a step-by-step breakdown, or do you have any questions?

Related Questions

1. How would the sequence change if the common ratio were negative?
2. What if we needed the 15th term instead?
3. How do we determine if a sequence is geometric based on its terms?
4. Could we find the sum of the first 10 terms in this geometric sequence?
5. How would the formula for the $n$-th term change if it were an arithmetic sequence?

Tip: Always verify if a sequence is geometric by calculating at least two ratios between consecutive terms.