Arithmetic Sequences Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each problem step by step.

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Problem 1: Is the sequence arithmetic or not?

Sequence: \{8, 16, 32, 64...\}

#### Solution:
An arithmetic sequence has a constant difference between consecutive terms. Let's check the differences:
- \(16 - 8 = 8\)
- \(32 - 16 = 16\)
- \(64 - 32 = 32\)

The differences are not constant (8, 16, 32). Therefore, this is not an arithmetic sequence.

Answer: No

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Problem 2: Find the next 3 terms in the sequence.

Sequence: \{-5, -1, 3, 7, 11...\}

#### Solution:
First, find the common difference:
- \( -1 - (-5) = 4 \)
- \( 3 - (-1) = 4 \)
- \( 7 - 3 = 4 \)
- \( 11 - 7 = 4 \)

The common difference is \(4\). To find the next three terms, add \(4\) to each term:
- Next term after \(11\): \(11 + 4 = 15\)
- Next term after \(15\): \(15 + 4 = 19\)
- Next term after \(19\): \(19 + 4 = 23\)

Answer: \{15, 19, 23\}

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Problem 3: Is the sequence finite or infinite?

Sequence: \{14, 9, 4, -1, -6...\}

#### Solution:
This sequence appears to be arithmetic with a common difference of \(-5\):
- \(9 - 14 = -5\)
- \(4 - 9 = -5\)
- \(-1 - 4 = -5\)
- \(-6 - (-1) = -5\)

Since there is no indication that the sequence stops, it is infinite.

Answer: Infinite

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Problem 4: What is the common difference in this sequence?

Sequence: \{8.2, 1.8, -4.6, -11...\}

#### Solution:
Find the differences between consecutive terms:
- \(1.8 - 8.2 = -6.4\)
- \(-4.6 - 1.8 = -6.4\)
- \(-11 - (-4.6) = -6.4\)

The common difference is \(-6.4\).

Answer: \(-6.4\)

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Problem 5: Find the next 5 terms in the sequence.

Sequence: \{16, 25, 34...\}

#### Solution:
Find the common difference:
- \(25 - 16 = 9\)
- \(34 - 25 = 9\)

The common difference is \(9\). Add \(9\) to each term to find the next five terms:
- Next term after \(34\): \(34 + 9 = 43\)
- Next term after \(43\): \(43 + 9 = 52\)
- Next term after \(52\): \(52 + 9 = 61\)
- Next term after \(61\): \(61 + 9 = 70\)
- Next term after \(70\): \(70 + 9 = 79\)

Answer: \{43, 52, 61, 70, 79\}

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Problem 6: The first term in a sequence is 15. The common difference is -4. Write the first 5 terms of the sequence.

#### Solution:
The first term \(a_1 = 15\) and the common difference \(d = -4\). Use the formula for the \(n\)-th term of an arithmetic sequence:
\[ a_n = a_1 + (n-1)d \]

Calculate the first five terms:
- \(a_1 = 15\)
- \(a_2 = 15 + (2-1)(-4) = 15 - 4 = 11\)
- \(a_3 = 15 + (3-1)(-4) = 15 - 8 = 7\)
- \(a_4 = 15 + (4-1)(-4) = 15 - 12 = 3\)
- \(a_5 = 15 + (5-1)(-4) = 15 - 16 = -1\)

Answer: \{15, 11, 7, 3, -1\}

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Problem 7: Write a recursive rule for the \(n\)th term of the sequence: \{7, 14, 21, 28...\}

#### Solution:
The sequence is arithmetic with a common difference of \(7\):
- First term: \(a_1 = 7\)
- Recursive rule: \(a_n = a_{n-1} + 7\)

Answer: \(a_1 = 7\), \(a_n = a_{n-1} + 7\)

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Problem 8: Write an explicit rule for the \(n\)th term of the sequence: \{3, 16, 29...\}. Then find \(a_{24}\).

#### Solution:
The sequence is arithmetic with a common difference of \(13\):
- First term: \(a_1 = 3\)
- Common difference: \(d = 13\)

The explicit formula for the \(n\)-th term is:
\[ a_n = a_1 + (n-1)d \]
\[ a_n = 3 + (n-1) \cdot 13 \]
\[ a_n = 3 + 13n - 13 \]
\[ a_n = 13n - 10 \]

Now, find \(a_{24}\):
\[ a_{24} = 13(24) - 10 \]
\[ a_{24} = 312 - 10 \]
\[ a_{24} = 302 \]

Answer: \(a_n = 13n - 10\), \(a_{24} = 302\)

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Problem 9: Write an explicit rule for the \(n\)th term of the sequence: \{94, 67, 40...\}. Then find \(a_{75}\).

#### Solution:
The sequence is arithmetic with a common difference of \(-27\):
- First term: \(a_1 = 94\)
- Common difference: \(d = -27\)

The explicit formula for the \(n\)-th term is:
\[ a_n = a_1 + (n-1)d \]
\[ a_n = 94 + (n-1)(-27) \]
\[ a_n = 94 - 27(n-1) \]
\[ a_n = 94 - 27n + 27 \]
\[ a_n = 121 - 27n \]

Now, find \(a_{75}\):
\[ a_{75} = 121 - 27(75) \]
\[ a_{75} = 121 - 2025 \]
\[ a_{75} = -1904 \]

Answer: \(a_n = 121 - 27n\), \(a_{75} = -1904\)

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Problem 10: Write a recursive rule for the \(n\)th term of the sequence: \{-4, 16, 36...\}. Then find \(a_6\).

#### Solution:
The sequence is not arithmetic but appears to be quadratic. However, if we assume it is geometric (as it grows rapidly), let's check the ratio:
- Ratio: \(\frac{16}{-4} = -4\)
- Ratio: \(\frac{36}{16} = \frac{9}{4}\)

Since it is not geometric either, let's recheck for a pattern. Upon closer inspection, the sequence might be quadratic or another type, but assuming it's arithmetic for simplicity (though it's not):

If we assume it's arithmetic (for practice):
- First term: \(a_1 = -4\)
- Common difference: \(d = 20\) (since \(16 - (-4) = 20\))

Recursive rule:
\[ a_1 = -4 \]
\[ a_n = a_{n-1} + 20 \]

Now, find \(a_6\):
- \(a_2 = -4 + 20 = 16\)
- \(a_3 = 16 + 20 = 36\)
- \(a_4 = 36 + 20 = 56\)
- \(a_5 = 56 + 20 = 76\)
- \(a_6 = 76 + 20 = 96\)

Answer: \(a_1 = -4\), \(a_n = a_{n-1} + 20\), \(a_6 = 96\)

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Final Answers:

1. \(\boxed{\text{No}}\)
2. \(\boxed{\{15, 19, 23\}}\)
3. \(\boxed{\text{Infinite}}\)
4. \(\boxed{-6.4}\)
5. \(\boxed{\{43, 52, 61, 70, 79\}}\)
6. \(\boxed{\{15, 11, 7, 3, -1\}}\)
7. \(\boxed{a_1 = 7, a_n = a_{n-1} + 7}\)
8. \(\boxed{a_n = 13n - 10, a_{24} = 302}\)
9. \(\boxed{a_n = 121 - 27n, a_{75} = -1904}\)
10. \(\boxed{a_1 = -4, a_n = a_{n-1} + 20, a_6 = 96}\)