a 
1
​
 =10 and a, start subscript, n, end subscript, equals, n, a, start subscript, n, minus, 1, end subscript, plus, 3a 
n
​
 =na 
n−1
​
 +3 then find the value of a, start subscript, 4, end subscripta 
4
​

We are given a recursive formula for \( a_n \), where the first term is \( a_1 = 10 \) and the recursive relationship is:

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

We need to find the value of \( a_4 \). Let’s calculate the sequence step by step using the recursive relation.

### Step 1: Find \( a_2 \)
Using the formula \( a_n = n a_{n-1} + 3 \) for \( n = 2 \):

\[
a_2 = 2a_1 + 3 = 2 \times 10 + 3 = 20 + 3 = 23
\]

### Step 2: Find \( a_3 \)
Using the same formula for \( n = 3 \):

\[
a_3 = 3a_2 + 3 = 3 \times 23 + 3 = 69 + 3 = 72
\]

### Step 3: Find \( a_4 \)
Finally, for \( n = 4 \):

\[
a_4 = 4a_3 + 3 = 4 \times 72 + 3 = 288 + 3 = 291
\]

Thus, the value of \( a_4 \) is:

\[
a_4 = 291
\]

Would you like more details or have any questions?

Here are 5 related questions you could explore:
1. How can we generalize a recursive sequence to a closed form?
2. What happens if we change the initial condition \( a_1 \)?
3. How can we compute higher-order terms like \( a_5 \) or \( a_6 \)?
4. How does the recursive formula impact the growth rate of the sequence?
5. Can you derive a non-recursive formula for \( a_n \) in this case?

**Tip:** When working with recursive sequences, try calculating a few terms manually to observe any patterns that could lead to shortcuts in finding the general term.

We are given a recursive formula for aₙ, where the first term is a₁ = 10 and the recursive relationship is: aₙ = n aₙ₋₁ + 3. Find the value of a₄.

Learn how to solve the recursive sequence problem aₙ = n aₙ₋₁ + 3 with a₁ = 10. This step-by-step solution explains how to calculate the value of a₄, covering core concepts like recursion and sequence progression. Suitable for students in Grades 9-11.

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