The image shows a list of sequences along with their recursive definitions, and you are asked to match each sequence with its corresponding recursive formula. Let's go through each sequence and the possible recursive relations:

1. $a_n = 6n$
   This sequence shows that each term is a multiple of 6. It follows the pattern where each term increases linearly by 6 with respect to $n$. This matches with recursive definition a) $a_{n+1} = a_n + 6$ with $a_1 = 6$.

2. $a_n = 6n$
   Same as the first one, this is again a linear sequence. Hence, it also matches with recursive relation a).

3. $a_n = 6n + 1$
   This sequence increases by 6 with an extra 1 added to each term. This fits with recursive definition c) $a_{n+1} = a_n + 6$ with $a_1 = 7$, which accounts for the constant 1 and starting point.

4. $a_n = 6^n$
   This sequence is exponential with a base of 6. The recursive relation that describes an exponential growth is d) $a_{n+1} = 6a_n$, with $a_1 = 6$.

Answer:

• 1. a
• 2. a
• 3. c
• 4. d

Would you like further explanation on any of these steps or need help with other questions?

Here are 5 related questions:

1. How do recursive sequences differ from explicit formulas?
2. Can you derive a recursive formula from an explicit sequence formula?
3. What are the benefits of using recursion in sequences?
4. How do we solve a recursive sequence with exponential growth?
5. How does the value of $a_1$ affect the behavior of the sequence?

Tip: Understanding the growth behavior of a sequence helps in identifying whether it's arithmetic, geometric, or exponential.