Given the recursively defined formula a = -3, aN = aN-1 + 4, determine which option is the correct function notation.

Answer 1

To analyze the recursively defined sequence given by $a_1=-3$ and $a_n=a_{n-1}+4$, we can determine the explicit formula for $a_n$.

1. Start with the recursive definition:
   • $a_1=-3$
   • $a_2=a_1+4=-3+4=1$
   • $a_3=a_2+4=1+4=5$
   • $a_4=a_3+4=5+4=9$
   • $a_5=a_4+4=9+4=13$

From these calculations, we can observe the pattern in the sequence:

• $a_1=-3$
• $a_2=1$
• $a_3=5$
• $a_4=9$
• $a_5=13$

Notice that the sequence is increasing by 4 each time.

2. We can determine the explicit formula for $a_n$ by recognizing that this sequence resembles an arithmetic sequence, where the first term is $-3$ and the common difference is $4$.

The formula for the $n$-th term of an arithmetic sequence can be expressed as: $$a_n=a_1+(n-1)d$$ Where $a_1$ is the first term and $d$ is the common difference.

Plugging our values into this formula:

• $a_1=-3$
• $d=4$

We have: $$a_n=-3+(n-1)\cdot 4$$

3. Simplifying this expression: $$a_n=-3+4n-4$$ $$a_n=4n-7$$

Thus, the correct function notation for the recursively defined sequence is: $$\boxed{{\displaystyle a_n=4n-7}}$$