https://file.aiquickdraw.com/mathbot/file/1725800196804wdywx26k.jpg

Let's solve the problems shown in the image.

### I. Find the first three terms of the sequence whose \(n\)-th term is given by the formula.

1. **\( a_n = n - 1 \)**
   \[
   a_1 = 1 - 1 = 0, \quad a_2 = 2 - 1 = 1, \quad a_3 = 3 - 1 = 2
   \]
   So, the first three terms are: \( 0, 1, 2 \).

2. **\( a_n = \frac{1}{n+1} \)**
   \[
   a_1 = \frac{1}{1+1} = \frac{1}{2}, \quad a_2 = \frac{1}{2+1} = \frac{1}{3}, \quad a_3 = \frac{1}{3+1} = \frac{1}{4}
   \]
   The first three terms are: \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \).

3. **\( a_n = \frac{4n}{n+2} \)**
   \[
   a_1 = \frac{4(1)}{1+2} = \frac{4}{3}, \quad a_2 = \frac{4(2)}{2+2} = \frac{8}{4} = 2, \quad a_3 = \frac{4(3)}{3+2} = \frac{12}{5}
   \]
   The first three terms are: \( \frac{4}{3}, 2, \frac{12}{5} \).

4. **\( a_n = 3n^2 + 4 \)**
   \[
   a_1 = 3(1)^2 + 4 = 3 + 4 = 7, \quad a_2 = 3(2)^2 + 4 = 12 + 4 = 16, \quad a_3 = 3(3)^2 + 4 = 27 + 4 = 31
   \]
   The first three terms are: \( 7, 16, 31 \).

5. **\( a_n = 2n - 3 \)**
   \[
   a_1 = 2(1) - 3 = 2 - 3 = -1, \quad a_2 = 2(2) - 3 = 4 - 3 = 1, \quad a_3 = 2(3) - 3 = 6 - 3 = 3
   \]
   The first three terms are: \( -1, 1, 3 \).

---

### II. Use the difference table to predict the next term in the sequence. Show your complete formula.

1. **\( 1, 6, 22, 56, 114, 202 \)**
   The differences between terms are:
   \[
   6 - 1 = 5, \quad 22 - 6 = 16, \quad 56 - 22 = 34, \quad 114 - 56 = 58, \quad 202 - 114 = 88
   \]
   The second differences:
   \[
   16 - 5 = 11, \quad 34 - 16 = 18, \quad 58 - 34 = 24, \quad 88 - 58 = 30
   \]
   Third differences are constant:
   \[
   18 - 11 = 7, \quad 24 - 18 = 6, \quad 30 - 24 = 6
   \]
   Since the third differences are nearly constant, this sequence is cubic. The next term can be predicted by continuing the third differences, leading to the next term: **326**.

2. **\( 9, 15, 21, 27, \dots \)**
   The differences are constant:
   \[
   15 - 9 = 6, \quad 21 - 15 = 6, \quad 27 - 21 = 6
   \]
   This is an arithmetic sequence with a common difference of 6. The next term is \( 27 + 6 = 33 \).

3. **\( 2, 6, 12, 20, 30, 42, \dots \)**
   The differences are:
   \[
   6 - 2 = 4, \quad 12 - 6 = 6, \quad 20 - 12 = 8, \quad 30 - 20 = 10, \quad 42 - 30 = 12
   \]
   The differences increase by 2, so the next difference will be 14, leading to the next term: \( 42 + 14 = 56 \).

4. **\( 6, 11, 18, 27, 38, \dots \)**
   The differences are:
   \[
   11 - 6 = 5, \quad 18 - 11 = 7, \quad 27 - 18 = 9, \quad 38 - 27 = 11
   \]
   The differences increase by 2, so the next difference will be 13, leading to the next term: \( 38 + 13 = 51 \).

5. **\( 1, 8, 21, 40, 65, 96, \dots \)**
   The differences are:
   \[
   8 - 1 = 7, \quad 21 - 8 = 13, \quad 40 - 21 = 19, \quad 65 - 40 = 25, \quad 96 - 65 = 31
   \]
   The differences increase by 6, so the next difference will be 37, leading to the next term: \( 96 + 37 = 133 \).

---

Let me know if you'd like further details or have any questions!

### Related Questions:
1. How do you determine if a sequence is arithmetic or geometric?
2. What are difference tables used for in sequences?
3. How do you find the formula for a quadratic sequence?
4. What is the general form of a cubic sequence formula?
5. How can you use the first differences to predict the next term in a sequence?

### Tip: 
For sequences that exhibit constant second or third differences, the sequence can often be modeled by a quadratic or cubic function respectively.

Find the first three terms of the sequence whose nth term is given by the formula. Use the difference table to predict the next term in the sequence.

Learn how to find the first three terms of a sequence based on given formulas, and how to predict the next term using a difference table. This problem covers sequences, algebraic expressions, and difference tables, suitable for Grades 9-11.

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