The first term of a sequence along with a recursion formula for the remaining terms is given below. Write out the first ten terms of the sequence.
a 1 equals 5 comma a Subscript n plus 1 Baseline equals StartFraction na Subscript n Over n plus 3 EndFraction

Given the first term \( a_1 = 5 \) and the recursion formula:

\[
a_{n+1} = \frac{n \cdot a_n}{n + 3}
\]

we can find the first ten terms of the sequence by calculating each term based on the previous one.

### Step-by-Step Calculation of Each Term
1. **First term**: \( a_1 = 5 \)
2. **Second term**: 
   \[
   a_2 = \frac{1 \cdot a_1}{1 + 3} = \frac{1 \cdot 5}{4} = \frac{5}{4} = 1.25
   \]
3. **Third term**: 
   \[
   a_3 = \frac{2 \cdot a_2}{2 + 3} = \frac{2 \cdot 1.25}{5} = \frac{2.5}{5} = 0.5
   \]
4. **Fourth term**: 
   \[
   a_4 = \frac{3 \cdot a_3}{3 + 3} = \frac{3 \cdot 0.5}{6} = \frac{1.5}{6} = 0.25
   \]
5. **Fifth term**: 
   \[
   a_5 = \frac{4 \cdot a_4}{4 + 3} = \frac{4 \cdot 0.25}{7} = \frac{1}{7} \approx 0.1429
   \]
6. **Sixth term**: 
   \[
   a_6 = \frac{5 \cdot a_5}{5 + 3} = \frac{5 \cdot 0.1429}{8} = \frac{0.7145}{8} \approx 0.0893
   \]
7. **Seventh term**: 
   \[
   a_7 = \frac{6 \cdot a_6}{6 + 3} = \frac{6 \cdot 0.0893}{9} = \frac{0.5358}{9} \approx 0.0595
   \]
8. **Eighth term**: 
   \[
   a_8 = \frac{7 \cdot a_7}{7 + 3} = \frac{7 \cdot 0.0595}{10} = \frac{0.4165}{10} \approx 0.0417
   \]
9. **Ninth term**: 
   \[
   a_9 = \frac{8 \cdot a_8}{8 + 3} = \frac{8 \cdot 0.0417}{11} = \frac{0.3336}{11} \approx 0.0303
   \]
10. **Tenth term**: 
    \[
    a_{10} = \frac{9 \cdot a_9}{9 + 3} = \frac{9 \cdot 0.0303}{12} = \frac{0.2727}{12} \approx 0.0227
    \]

### First Ten Terms of the Sequence
Thus, the first ten terms are approximately:
\[
5, 1.25, 0.5, 0.25, 0.1429, 0.0893, 0.0595, 0.0417, 0.0303, 0.0227
\]

Would you like more details or have any questions?

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**Related Questions:**
1. How does the recursion formula affect the behavior of this sequence as \( n \) increases?
2. What is the general behavior of the sequence in terms of convergence?
3. How would the sequence change if the initial term \( a_1 \) were different?
4. Can the recursive formula be expressed as a closed-form expression?
5. How can this type of sequence be applied in real-world contexts?

**Tip:** Recurrence relations often lead to sequences that either converge, oscillate, or grow. Analyzing the recursion formula can give insights into the behavior of the sequence.

The first term of a sequence along with a recursion formula for the remaining terms is given below. Write out the first ten terms of the sequence. a1 = 5, an+1 = (n * an) / (n + 3)

Explore how to determine the first ten terms of a recursive sequence with an initial term a1 = 5 and a recurrence formula an+1 = (n * an) / (n + 3). This solution covers key mathematical concepts in sequences and recurrence relations, with step-by-step calculations for each term. Suitable for high school students in Grades 9-12.

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