The problem in the image asks you to match a recursive definition to a sequence based on the given conditions.

Let's break down each sequence:

a) $a_{n+1} = a_n + 6$ for $n \geq 1$ and $a_1 = 6$

This is an arithmetic sequence where each term is 6 more than the previous one, starting with $a_1 = 6$.

b) $a_{n+1} = a_n$ for $n \geq 1$ and $a_1 = 6$

This is a constant sequence where all terms are equal to 6.

c) $a_{n+1} = a_n + 6$ for $n \geq 1$ and $a_1 = 7$

This is another arithmetic sequence, similar to the first one, but it starts with 7 instead of 6.

d) $a_{n+1} = 6a_n$ for $n \geq 1$ and $a_1 = 6$

This is a geometric sequence where each term is 6 times the previous one, starting with $a_1 = 6$.

Now, let's match the recursive formulas to the sequences described in the problem:

• b) $a_{n+1} = a_n + 6$ for $n \geq 1$ and $a_1 = 6$ matches the sequence $a_n = 6n$.
• a) $a_{n+1} = a_n + 6$ for $n \geq 1$ and $a_1 = 6$ matches the sequence $a_n = 6n + 1$.
• c) $a_{n+1} = a_n + 6$ for $n \geq 1$ and $a_1 = 7$ matches the sequence $a_n = 6n + 7$.
• d) $a_{n+1} = 6a_n$ for $n \geq 1$ and $a_1 = 6$ matches the sequence $a_n = 6^n$.

So the answers are:

• a) corresponds to 4
• b) corresponds to 3
• c) corresponds to 2
• d) corresponds to 1

Let me know if you'd like any further details or have questions! Here are some related questions you can explore:

1. How do you identify whether a sequence is arithmetic or geometric?
2. Can a sequence have both recursive and explicit formulas?
3. What is the difference between recursive and closed-form expressions in sequences?
4. How do you prove that a sequence follows a recursive definition?
5. Can sequences with different starting points have the same recursive form?

Tip: When solving sequences, always check whether the sequence involves addition (arithmetic) or multiplication (geometric), as that will guide the recursive definition.