 the first two terms of the sequence given the nth term.
1. an = (-2)^2n-1
2. an = n - 1/n
a. Write the 1st 4 terms of the sequence whose nth term is given by the formula.
a = 3n - 1
b. Find the next two terms of each quence: 15, 7, -1, -9,
c. Find the indicated term of the sequence whose nth term is given by the formula.
an = (n+2)(n+3) , n17

Let's solve each part of the problem step by step.

### 1. Find the first two terms of the given sequences:
#### (a) Sequence: \( a_n = (-2)^{2n-1} \)

For \( n = 1 \):
\[ a_1 = (-2)^{2(1)-1} = (-2)^1 = -2 \]

For \( n = 2 \):
\[ a_2 = (-2)^{2(2)-1} = (-2)^3 = -8 \]

So, the first two terms are \(-2\) and \(-8\).

#### (b) Sequence: \( a_n = n - \frac{1}{n} \)

For \( n = 1 \):
\[ a_1 = 1 - \frac{1}{1} = 1 - 1 = 0 \]

For \( n = 2 \):
\[ a_2 = 2 - \frac{1}{2} = 2 - 0.5 = 1.5 \]

So, the first two terms are \(0\) and \(1.5\).

### 2. Write the first 4 terms of the sequence whose nth term is given by the formula \( a_n = 3n - 1 \).

For \( n = 1 \):
\[ a_1 = 3(1) - 1 = 3 - 1 = 2 \]

For \( n = 2 \):
\[ a_2 = 3(2) - 1 = 6 - 1 = 5 \]

For \( n = 3 \):
\[ a_3 = 3(3) - 1 = 9 - 1 = 8 \]

For \( n = 4 \):
\[ a_4 = 3(4) - 1 = 12 - 1 = 11 \]

So, the first 4 terms are \( 2, 5, 8, 11 \).

### 3. Find the next two terms of the sequence: \( 15, 7, -1, -9 \).

To find the pattern, observe the differences between consecutive terms:

\[ 7 - 15 = -8 \]
\[ -1 - 7 = -8 \]
\[ -9 - (-1) = -8 \]

The common difference is \(-8\).

Next term after \(-9\):
\[ -9 - 8 = -17 \]

Next term after \(-17\):
\[ -17 - 8 = -25 \]

So, the next two terms are \(-17\) and \(-25\).

### 4. Find the indicated term of the sequence whose nth term is given by the formula \( a_n = (n+2)(n+3) \).

For \( n = 17 \):
\[ a_{17} = (17+2)(17+3) = 19 \times 20 = 380 \]

So, the 17th term is \( 380 \).

### Summary:
1. First two terms of \( a_n = (-2)^{2n-1} \): \(-2, -8\).
2. First two terms of \( a_n = n - \frac{1}{n} \): \( 0, 1.5 \).
3. First 4 terms of \( a_n = 3n - 1 \): \( 2, 5, 8, 11 \).
4. Next two terms of \( 15, 7, -1, -9 \): \(-17, -25 \).
5. 17th term of \( a_n = (n+2)(n+3) \): \( 380 \).

Would you like further details or have any questions?

Here are 8 related questions to expand on this information:

1. What is the 3rd term of the sequence \( a_n = (-2)^{2n-1} \)?
2. What is the 3rd term of the sequence \( a_n = n - \frac{1}{n} \)?
3. What is the 5th term of the sequence \( a_n = 3n - 1 \)?
4. Can you identify the common difference in an arithmetic sequence?
5. How do you determine the nth term in a geometric sequence?
6. What is the formula for the sum of the first n terms of an arithmetic sequence?
7. How do you find the nth term of a sequence if the common ratio is given?
8. What are some real-life applications of sequences and series?

**Tip:** When finding terms of a sequence, identify the pattern (difference or ratio) to simplify the calculation of subsequent terms.

Explore solutions to sequences problems including finding first two terms, next two terms, and the indicated term. Covers arithmetic and geometric sequences with detailed step-by-step solutions. Suitable for high school students.

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