Sure, let's work on the given problem step-by-step.
### Step-by-Step Solution
The given number pattern is:
[tex]\[ 1, 5, 9, \ldots \][/tex]
#### Step 1: Determine the Pattern
To find the pattern, let's look at the differences between consecutive terms:
- The difference between the second term (5) and the first term (1) is [tex]\( 5 - 1 = 4 \)[/tex].
- The difference between the third term (9) and the second term (5) is [tex]\( 9 - 5 = 4 \)[/tex].
We see that the difference between consecutive terms is constant and equal to 4. This type of pattern is known as an arithmetic sequence where the common difference (d) is 4.
#### Step 2: Find the Next Three Terms
Knowing the common difference (d = 4), we can find the next three terms by adding this common difference to the last known term repeatedly.
1. The last known term is 9:
- Next term: [tex]\( 9 + 4 = 13 \)[/tex]
2. Then, using 13:
- Next term: [tex]\( 13 + 4 = 17 \)[/tex]
3. Then, using 17:
- Next term: [tex]\( 17 + 4 = 21 \)[/tex]
So, the next three terms in the pattern are:
[tex]\[ 13, 17, 21 \][/tex]
### Final Answer
(a) The next three terms in the sequence are:
[tex]\[ 13, 17, 21 \][/tex]
### Explanation in Words
The number pattern given is an arithmetic sequence where each term increases by a constant difference of 4. Starting from the initial terms 1, 5, and 9, the subsequent terms are determined by adding 4 to the previous term. Thus, the pattern continues with the terms 13, 17, and 21 following the same rule.
