find the tenth term in each sequence.
9) $-1, \frac{2}{3}, \frac{7}{3}, 4, \frac{17}{3},...$ skip
10) $7, 9, 12, 16, 21,...$
11) $-2, -6, -18, -54, -162,...$
12) $-23, -18, -13, -8, -3,...$
13) $-4, 12, -36, 108, -324,...$
extra credit 14) $-6, -2, 0, 1, \frac{3}{2},...$
15) $-28, 172, 372, 572, 772,...$
16) $37, 46, 55, 64, 73,...$
find the first four terms in each sequence.
17) $a_n = \frac{2n + 1}{n^3}$
18) $a_n = 3^{n - 1}$
19) $a_n = n^2 + 1$
20) $a_n = \frac{n^3}{2n + 1}$

Question

Accepted Answer

Let's solve some of these sequence problems. Let's start with problem 10: Find the tenth term of the sequence $ 7, 9, 12, 16, 21, \dots $

# Explanation:
## Step 1: Identify the pattern (differences between terms)
First, find the differences between consecutive terms:
- $ 9 - 7 = 2 $
- $ 12 - 9 = 3 $
- $ 16 - 12 = 4 $
- $ 21 - 16 = 5 $

So the differences are increasing by 1 each time: $ 2, 3, 4, 5, \dots $

## Step 2: Find the nth term formula
The nth term of the sequence can be found by summing the initial term and the sum of the differences. The initial term $ a_1 = 7 $. The differences form an arithmetic sequence with first term $ d_1 = 2 $ and common difference $ 1 $.

The sum of the first $ (n - 1) $ differences is the sum of an arithmetic series: $ S_{n-1} = \frac{(n - 1)}{2} \times [2 \times 2 + (n - 2) \times 1] $

Simplify:
$ S_{n-1} = \frac{(n - 1)}{2} \times (4 + n - 2) = \frac{(n - 1)(n + 2)}{2} $

Then the nth term $ a_n = 7 + \frac{(n - 1)(n + 2)}{2} $

## Step 3: Calculate the 10th term
Substitute $ n = 10 $ into the formula:
$ a_{10} = 7 + \frac{(10 - 1)(10 + 2)}{2} $
$ a_{10} = 7 + \frac{9 \times 12}{2} $
$ a_{10} = 7 + \frac{108}{2} $
$ a_{10} = 7 + 54 $
$ a_{10} = 61 $ (Wait, but let's check the handwritten note: 27, 34, 42, 51, 61. Wait, maybe another way. Let's list the terms:

Term 1: 7

Term 2: 7 + 2 = 9

Term 3: 9 + 3 = 12

Term 4: 12 + 4 = 16

Term 5: 16 + 5 = 21

Term 6: 21 + 6 = 27

Term 7: 27 + 7 = 34

Term 8: 34 + 8 = 42

Term 9: 42 + 9 = 51

Term 10: 51 + 10 = 61

Yes, that matches. So the 10th term is 61.

Now problem 11: $ -2, -6, -18, -54, -162, \dots $

# Explanation:
## Step 1: Identify the pattern (common ratio)
Check the ratio between consecutive terms:
- $ \frac{-6}{-2} = 3 $
- $ \frac{-18}{-6} = 3 $
- $ \frac{-54}{-18} = 3 $
- $ \frac{-162}{-54} = 3 $

So it's a geometric sequence with first term $ a_1 = -2 $ and common ratio $ r = 3 $

## Step 2: Geometric sequence formula
The nth term of a geometric sequence is $ a_n = a_1 \times r^{n - 1} $

## Step 3: Calculate the 10th term
Substitute $ n = 10 $, $ a_1 = -2 $, $ r = 3 $:
$ a_{10} = -2 \times 3^{10 - 1} $
$ a_{10} = -2 \times 3^9 $
$ 3^9 = 19683 $
$ a_{10} = -2 \times 19683 = -39366 $

Problem 12: $ -23, -18, -13, -8, -3, \dots $

# Explanation:
## Step 1: Identify the pattern (common difference)
Check the difference between consecutive terms:
- $ -18 - (-23) = 5 $
- $ -13 - (-18) = 5 $
- $ -8 - (-13) = 5 $
- $ -3 - (-8) = 5 $

So it's an arithmetic sequence with first term $ a_1 = -23 $ and common difference $ d = 5 $

## Step 2: Arithmetic sequence formula
The nth term of an arithmetic sequence is $ a_n = a_1 + (n - 1)d $

## Step 3: Calculate the 10th term
Substitute $ n = 10 $, $ a_1 = -23 $, $ d = 5 $:
$ a_{10} = -23 + (10 - 1) \times 5 $
$ a_{10} = -23 + 9 \times 5 $
$ a_{10} = -23 + 45 = 22 $

Problem 13: $ -4, 12, -36, 108, -324, \dots $

# Explanation:
## Step 1: Identify the pattern (common ratio)
Check the ratio between consecutive terms:
- $ \frac{12}{-4} = -3 $
- $ \frac{-36}{12} = -3 $
- $ \frac{108}{-36} = -3 $
- $ \frac{-324}{108} = -3 $

So it's a geometric sequence with first term $ a_1 = -4 $ and common ratio $ r = -3 $

## Step 2: Geometric sequence formula
The nth term is $ a_n = a_1 \times r^{n - 1} $

## Step 3: Calculate the 10th term
Substitute $ n = 10 $, $ a_1 = -4 $, $ r = -3 $:
$ a_{10} = -4 \times (-3)^{10 - 1} $
$ a_{10} = -4 \times (-3)^9 $
$ (-3)^9 = -19683 $
$ a_{10} = -4 \times (-19683) = 78732 $

Problem 14: $ -6, -2, 0, 1, \frac{3}{2}, \dots $

# Explanation:
## Step 1: Identify the pattern (differences between terms)
First, find the differences between consecutive terms:
- $ -2 - (-6) = 4 $
- $ 0 - (-2) = 2 $
- $ 1 - 0 = 1 $
- $ \frac{3}{2} - 1 = \frac{1}{2} $

The differences are $ 4, 2, 1, \frac{1}{2}, \dots $, which is a geometric sequence with first term 4 and common ratio $ \frac{1}{2} $

## Step 2: Find the nth term formula
The nth term $ a_n $ can be expressed as the initial term plus the sum of the geometric series of differences.

The initial term $ a_1 = -6 $

The sum of the first $ (n - 1) $ differences (geometric series) is $ S_{n - 1} = 4 \times \frac{1 - (\frac{1}{2})^{n - 1}}{1 - \frac{1}{2}} = 4 \times \frac{1 - (\frac{1}{2})^{n - 1}}{\frac{1}{2}} = 8 \times (1 - (\frac{1}{2})^{n - 1}) $

So $ a_n = -6 + 8 \times (1 - (\frac{1}{2})^{n - 1}) $

Simplify: $ a_n = -6 + 8 - 8 \times (\frac{1}{2})^{n - 1} = 2 - \frac{8}{2^{n - 1}} = 2 - \frac{2^3}{2^{n - 1}} = 2 - 2^{4 - n} $

## Step 3: Calculate the 10th term
Substitute $ n = 10 $:
$ a_{10} = 2 - 2^{4 - 10} = 2 - 2^{-6} = 2 - \frac{1}{64} = \frac{128}{64} - \frac{1}{64} = \frac{127}{64} = 1.984375 $

Wait, let's check by listing terms:

Term 1: -6

Term 2: -6 + 4 = -2

Term 3: -2 + 2 = 0

Term 4: 0 + 1 = 1

Term 5: 1 + 0.5 = 1.5 = 3/2

Term 6: 1.5 + 0.25 = 1.75

Term 7: 1.75 + 0.125 = 1.875

Term 8: 1.875 + 0.0625 = 1.9375

Term 9: 1.9375 + 0.03125 = 1.96875

Term 10: 1.96875 + 0.015625 = 1.984375 = 127/64

Yes, that matches.

Problem 15: $ -28, 172, 372, 572, 772, \dots $

# Explanation:
## Step 1: Identify the pattern (common difference)
Check the difference between consecutive terms:
- $ 172 - (-28) = 200 $
- $ 372 - 172 = 200 $
- $ 572 - 372 = 200 $
- $ 772 - 572 = 200 $

So it's an arithmetic sequence with first term $ a_1 = -28 $ and common difference $ d = 200 $

## Step 2: Arithmetic sequence formula
$ a_n = a_1 + (n - 1)d $

## Step 3: Calculate the 10th term
Substitute $ n = 10 $, $ a_1 = -28 $, $ d = 200 $:
$ a_{10} = -28 + (10 - 1) \times 200 $
$ a_{10} = -28 + 9 \times 200 $
$ a_{10} = -28 + 1800 = 1772 $

Problem 16: $ 37, 46, 55, 64, 73, \dots $

# Explanation:
## Step 1: Identify the pattern (common difference)
Check the difference between consecutive terms:
- $ 46 - 37 = 9 $
- $ 55 - 46 = 9 $
- $ 64 - 55 = 9 $
- $ 73 - 64 = 9 $

So it's an arithmetic sequence with first term $ a_1 = 37 $ and common difference $ d = 9 $

## Step 2: Arithmetic sequence formula
$ a_n = a_1 + (n - 1)d $

## Step 3: Calculate the 10th term
Substitute $ n = 10 $, $ a_1 = 37 $, $ d = 9 $:
$ a_{10} = 37 + (10 - 1) \times 9 $
$ a_{10} = 37 + 9 \times 9 $
$ a_{10} = 37 + 81 = 118 $

Problem 18: $ a_n = 3^{n - 1} $, find the first four terms.

# Explanation:
## Step 1: Calculate $ a_1 $
Substitute $ n = 1 $: $ a_1 = 3^{1 - 1} = 3^0 = 1 $

## Step 2: Calculate $ a_2 $
Substitute $ n = 2 $: $ a_2 = 3^{2 - 1} = 3^1 = 3 $

## Step 3: Calculate $ a_3 $
Substitute $ n = 3 $: $ a_3 = 3^{3 - 1} = 3^2 = 9 $

## Step 4: Calculate $ a_4 $
Substitute $ n = 4 $: $ a_4 = 3^{4 - 1} = 3^3 = 27 $

Problem 19: $ a_n = n^2 + 1 $, find the first four terms.

# Explanation:
## Step 1: Calculate $ a_1 $
Substitute $ n = 1 $: $ a_1 = 1^2 + 1 = 1 + 1 = 2 $

## Step 2: Calculate $ a_2 $
Substitute $ n = 2 $: $ a_2 = 2^2 + 1 = 4 + 1 = 5 $

## Step 3: Calculate $ a_3 $
Substitute $ n = 3 $: $ a_3 = 3^2 + 1 = 9 + 1 = 10 $

## Step 4: Calculate $ a_4 $
Substitute $ n = 4 $: $ a_4 = 4^2 + 1 = 16 + 1 = 17 $

Problem 20: $ a_n = \frac{n^3}{2n + 1} $, find the first four terms.

# Explanation:
## Step 1: Calculate $ a_1 $
Substitute $ n = 1 $: $ a_1 = \frac{1^3}{2(1) + 1} = \frac{1}{2 + 1} = \frac{1}{3} $

## Step 2: Calculate $ a_2 $
Substitute $ n = 2 $: $ a_2 = \frac{2^3}{2(2) + 1} = \frac{8}{4 + 1} = \frac{8}{5} $

## Step 3: Calculate $ a_3 $
Substitute $ n = 3 $: $ a_3 = \frac{3^3}{2(3) + 1} = \frac{27}{6 + 1} = \frac{27}{7} $

## Step 4: Calculate $ a_4 $
Substitute $ n = 4 $: $ a_4 = \frac{4^3}{2(4) + 1} = \frac{64}{8 + 1} = \frac{64}{9} $

# Answers:
10) $ \boldsymbol{61} $

11) $ \boldsymbol{-39366} $

12) $ \boldsymbol{22} $

13) $ \boldsymbol{78732} $

14) $ \boldsymbol{\frac{127}{64}} $ (or 1.984375)

15) $ \boldsymbol{1772} $

16) $ \boldsymbol{118} $

18) First four terms: $ \boldsymbol{1, 3, 9, 27} $

19) First four terms: $ \boldsymbol{2, 5, 10, 17} $

20) First four terms: $ \boldsymbol{\frac{1}{3}, \frac{8}{5}, \frac{27}{7}, \frac{64}{9}} $

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QUESTION IMAGE

Question

Explanation:

Step 1: Identify the pattern (differences between terms)

Step 2: Find the nth term formula

Step 3: Calculate the 10th term

Step 1: Identify the pattern (common ratio)

Step 2: Geometric sequence formula

Step 3: Calculate the 10th term

Step 1: Identify the pattern (common difference)

Step 2: Arithmetic sequence formula

Step 3: Calculate the 10th term

Answer: