Arithmetic Sequences Notes and Worksheets - Lindsay Bowden - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

1. Is the sequence arithmetic or not? $\{8, 16, 32, 64...\}$
* Check: To be an arithmetic sequence, you must add the same number to get from one term to the next.
* $16 - 8 = 8$
* $32 - 16 = 16$
* Reasoning: Since we added 8 the first time and 16 the second time, the difference is not constant. (This is actually a geometric sequence where you multiply by 2).
* Result: Not arithmetic.

2. Find the next 3 terms in the sequence. $\{-5, -1, 3, 7, 11...\}$
* Find the pattern: Look at the difference between numbers.
* $-1 - (-5) = 4$
* $3 - (-1) = 4$
* The common difference is $+4$.
* Calculate next terms:
* $11 + 4 = 15$
* $15 + 4 = 19$
* $19 + 4 = 23$

3. Is the sequence finite or infinite? $\{14, 9, 4, -1, -6...\}$
* Check: Look at the end of the list.
* Reasoning: The ellipsis (...) at the end indicates that the sequence continues forever without stopping.
* Result: Infinite.

4. What is the common difference in this sequence? $\{8.2, 1.8, -4.6, -11\}$
* Calculate: Subtract the first term from the second term.
* $1.8 - 8.2 = -6.4$
* Verify: Check the next pair to be sure.
* $-4.6 - 1.8 = -6.4$
* Result: $-6.4$

5. Find the next 5 terms in the sequence. $\{16, 25, 34...\}$
* Find the pattern:
* $25 - 16 = 9$
* $34 - 25 = 9$
* The rule is add 9.
* Calculate next terms:
* $34 + 9 = 43$
* $43 + 9 = 52$
* $52 + 9 = 61$
* $61 + 9 = 70$
* $70 + 9 = 79$

6. The first term in a sequence is 15. The common difference is -4. Write the first 5 terms of the sequence.
* Start: 15
* Rule: Subtract 4 each time.
* Term 1: $15$
* Term 2: $15 - 4 = 11$
* Term 3: $11 - 4 = 7$
* Term 4: $7 - 4 = 3$
* Term 5: $3 - 4 = -1$

7. Write a recursive rule for the $n$th term of the sequence: $\{7, 14, 21, 28...\}$
* Identify parts:
* First term ($a_1$) = 7
* Common difference ($d$) = $14 - 7 = 7$
* Format: A recursive rule needs the starting point and the formula for the next term based on the previous one ($a_{n-1}$).
* Result: $a_1 = 7; \quad a_n = a_{n-1} + 7$

8. Write an explicit rule for the $n$th term of the sequence: $\{3, 16, 29...\}$. Then find $a_{24}$.
* Identify parts:
* First term ($a_1$) = 3
* Common difference ($d$) = $16 - 3 = 13$
* Formula: $a_n = a_1 + (n-1)d$
* Write Rule: $a_n = 3 + (n-1)13$
* Simplify: $a_n = 3 + 13n - 13 \rightarrow a_n = 13n - 10$
* Find $a_{24}$: Plug in 24 for $n$.
* $a_{24} = 13(24) - 10$
* $13 \times 24 = 312$
* $312 - 10 = 302$

9. Write an explicit rule for the $n$th term of the sequence: $\{94, 67, 40...\}$. Then find $a_{75}$.
* Identify parts:
* First term ($a_1$) = 94
* Common difference ($d$) = $67 - 94 = -27$
* Formula: $a_n = a_1 + (n-1)d$
* Write Rule: $a_n = 94 + (n-1)(-27)$
* Simplify: $a_n = 94 - 27n + 27 \rightarrow a_n = -27n + 121$
* Find $a_{75}$: Plug in 75 for $n$.
* $a_{75} = -27(75) + 121$
* $-27 \times 75 = -2025$
* $-2025 + 121 = -1904$

10. Write a recursive rule for the $n$th term of the sequence: $\{-4, 16, 36...\}$. Then find $a_6$.
* Identify parts:
* First term ($a_1$) = -4
* Common difference ($d$) = $16 - (-4) = 20$
* Write Recursive Rule:
* $a_1 = -4; \quad a_n = a_{n-1} + 20$
* Find $a_6$: You can count up step-by-step using the recursive rule.
* $a_1 = -4$
* $a_2 = 16$
* $a_3 = 36$
* $a_4 = 36 + 20 = 56$
* $a_5 = 56 + 20 = 76$
* $a_6 = 76 + 20 = 96$

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Final Answer:
1. Not arithmetic
2. 15, 19, 23
3. Infinite
4. -6.4
5. 43, 52, 61, 70, 79
6. 15, 11, 7, 3, -1
7. $a_1 = 7; a_n = a_{n-1} + 7$
8. Rule: $a_n = 13n - 10$; $a_{24} = 302$
9. Rule: $a_n = -27n + 121$; $a_{75} = -1904$
10. Rule: $a_1 = -4; a_n = a_{n-1} + 20$; $a_6 = 96$