To solve for $k$ in this sequence, we can use the given term-to-term rule: "Multiply by 2 then add $k$." Let's break it down.

Step 1: Understanding the problem

• The first term is $3$.
• The last term provided is $87$.
• The rule is: $\text{Next term} = (\text{Current term} \times 2) + k$

Step 2: Form equations for the sequence

1. First term: $3$ $\text{Second term} = (3 \times 2) + k = 6 + k$

2. Second term leads to the third term: $\text{Third term} = ((6 + k) \times 2) + k = 12 + 2k + k = 12 + 3k$

3. Third term leads to the fourth term, which is $87$: $((12 + 3k) \times 2) + k = 87$ Simplify: $24 + 6k + k = 87$ $24 + 7k = 87$

Step 3: Solve for $k$

$7k = 87 - 24$ $7k = 63$ $k = 9$

Final Answer:

The value of $k$ is $9$.

Would you like further clarification on this problem or assistance with similar sequences? Here are 5 related questions to consider:

1. How does the term-to-term rule work in other types of sequences?
2. Can you derive the explicit formula for the $n$-th term of this sequence?
3. What if a sequence has multiple unknowns in its rule? How do you solve it?
4. How can we verify that $k = 9$ is correct?
5. Can this rule be applied in reverse to find previous terms?

Tip: Always verify your result by substituting $k$ back into the sequence!