The question on the image asks to create a new problem using a pattern in a sequence of numbers. It provides an example where the sequence is $1, 4, 7, \ldots$, and it asks for the 7th term in this sequence.

Solution:

The given sequence $1, 4, 7, \ldots$ is an arithmetic sequence where each term increases by 3. The formula for the $n$-th term of an arithmetic sequence is:

$a_n = a_1 + (n - 1) \cdot d$

where:

• $a_1$ is the first term,
• $d$ is the common difference, and
• $n$ is the term number.

For this sequence:

• $a_1 = 1$ (the first term),
• $d = 3$ (the common difference),
• $n = 7$ (we need to find the 7th term).

Plugging in the values:

$a_7 = 1 + (7 - 1) \cdot 3$

$a_7 = 1 + 6 \cdot 3$

$a_7 = 1 + 18$

$a_7 = 19$

Answer:

The 7th term in the sequence is 19.

Would you like a more detailed explanation or have any other questions about this?

Here are 5 related questions:

1. How do you find the common difference in an arithmetic sequence?
2. What is the formula to find the sum of the first $n$ terms of an arithmetic sequence?
3. How can we determine if a sequence is arithmetic or geometric?
4. How do you derive the formula for the $n$-th term of an arithmetic sequence?
5. Can you explain how to solve for $n$ if you are given a term value in an arithmetic sequence?

Tip: Always check the common difference first when identifying the pattern in an arithmetic sequence. It helps in quickly determining the sequence's rule.