Math 10 worksheet on sequences and series with activities to classify, complete, and generate patterns.

Problem Analysis and Solution

The worksheet focuses on sequences and series, which are fundamental concepts in mathematics. Let's solve each activity step by step.

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#### Activity 1: FILLED ME UP!!!
Task: Classify whether the given is a sequence or a series. Write the letter of the correct answer in the box provided.

- Sequence: A list of numbers in a specific order.
- Series: The sum of the terms of a sequence.

##### Given options:
1. A. \(2, 4, 6, 8, 10\)
- This is a list of numbers in a specific order (even numbers).
- Classification: Sequence

2. B. \(3 + 6 + 9 + 12 = 30\)
- This is the sum of the terms of a sequence.
- Classification: Series

3. C. The sum of all the elements is 90.
- This refers to the total sum of elements, indicating a series.
- Classification: Series

4. D. The elements of a set are \(1, 2, 4, 8, 16\).
- This is a list of numbers in a specific order (powers of 2).
- Classification: Sequence

##### Final Answer for Activity 1:
- SEQUENCE: A, D
- SERIES: B, C

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#### Activity 2: COMPLETE ME!!!
Task: Complete the table by identifying the first and last terms, and determining whether the given is a finite or infinite sequence.

##### Table to complete:
| NO. | SEQUENCE | FIRST TERM | LAST TERM | INFINITE OR FINITE |
|------|-----------------------------------|------------|-----------|--------------------|
| 1 | \(\ldots, -64, -49, -36, -25, -16\) | Extended | | |
| 2 | \(1, 2, 3, 4, 5\) | | 5 | |
| 3 | \(-9, -2, 5, 12, 19, \ldots\) | | | Infinite |
| 4 | \(-2, 4, -8, \ldots, -2048, 4096\) | -2 | | |
| 5 | \(38, 33, 28, 23, 18, 13\) | | 13 | |

##### Step-by-step analysis:

1. Row 1: \(\ldots, -64, -49, -36, -25, -16\)
- The sequence appears to be decreasing squares of integers: \(-8^2, -7^2, -6^2, -5^2, -4^2\).
- First Term: Extended (not specified).
- Last Term: Not specified (infinite sequence).
- Finite or Infinite: Infinite.

2. Row 2: \(1, 2, 3, 4, 5\)
- The sequence is the first five positive integers.
- First Term: 1
- Last Term: 5
- Finite or Infinite: Finite.

3. Row 3: \(-9, -2, 5, 12, 19, \ldots\)
- The sequence increases by 7 each time: \(-9, -2, 5, 12, 19, \ldots\).
- First Term: -9
- Last Term: Not specified (infinite sequence).
- Finite or Infinite: Infinite.

4. Row 4: \(-2, 4, -8, \ldots, -2048, 4096\)
- The sequence alternates signs and doubles each time: \((-2)^1, (-2)^2, (-2)^3, \ldots\).
- First Term: -2
- Last Term: 4096
- Finite or Infinite: Finite.

5. Row 5: \(38, 33, 28, 23, 18, 13\)
- The sequence decreases by 5 each time: \(38, 33, 28, 23, 18, 13\).
- First Term: 38
- Last Term: 13
- Finite or Infinite: Finite.

##### Final Answer for Activity 2:
| NO. | SEQUENCE | FIRST TERM | LAST TERM | INFINITE OR FINITE |
|------|-----------------------------------|------------|-----------|--------------------|
| 1 | \(\ldots, -64, -49, -36, -25, -16\) | Extended | N/A | Infinite |
| 2 | \(1, 2, 3, 4, 5\) | 1 | 5 | Finite |
| 3 | \(-9, -2, 5, 12, 19, \ldots\) | -9 | N/A | Infinite |
| 4 | \(-2, 4, -8, \ldots, -2048, 4096\) | -2 | 4096 | Finite |
| 5 | \(38, 33, 28, 23, 18, 13\) | 38 | 13 | Finite |

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#### Activity 3: MISSING U!!!
Task: Find the missing number after the given sequences.

##### Sequences to analyze:
1. \(21, 22, 43, 67, \_\)
- Pattern: Each term is the sum of the previous two terms.
- \(21 + 22 = 43\)
- \(22 + 43 = 67\)
- Next term: \(43 + 67 = 110\)
- Missing Number: 110

2. \(16, 8, 4, 2, 1, \_\)
- Pattern: Each term is half of the previous term.
- \(16 \div 2 = 8\)
- \(8 \div 2 = 4\)
- \(4 \div 2 = 2\)
- \(2 \div 2 = 1\)
- Next term: \(1 \div 2 = 0.5\)
- Missing Number: 0.5

3. \(13, 25, 47, 69, \_\)
- Pattern: Each term increases by 12, then 22, then 22, and so on.
- \(25 - 13 = 12\)
- \(47 - 25 = 22\)
- \(69 - 47 = 22\)
- Next term: \(69 + 22 = 91\)
- Missing Number: 91

4. \(2, 3, 5, 7, 11, \_\)
- Pattern: Prime numbers in ascending order.
- Next prime number after 11 is 13.
- Missing Number: 13

5. \(\frac{1}{2}, \frac{3}{5}, \frac{5}{8}, \frac{7}{11}, \_\)
- Pattern: Numerator increases by 2 each time (\(1, 3, 5, 7, \ldots\)).
- Denominator increases by 3 each time (\(2, 5, 8, 11, \ldots\)).
- Next term: \(\frac{9}{14}\)
- Missing Number: \(\frac{9}{14}\)

##### Final Answer for Activity 3:
1. 110
2. 0.5
3. 91
4. 13
5. \(\frac{9}{14}\)

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Final Answers:

- Activity 1:
- SEQUENCE: A, D
- SERIES: B, C

- Activity 2:
| NO. | SEQUENCE | FIRST TERM | LAST TERM | INFINITE OR FINITE |
|------|-----------------------------------|------------|-----------|--------------------|
| 1 | \(\ldots, -64, -49, -36, -25, -16\) | Extended | N/A | Infinite |
| 2 | \(1, 2, 3, 4, 5\) | 1 | 5 | Finite |
| 3 | \(-9, -2, 5, 12, 19, \ldots\) | -9 | N/A | Infinite |
| 4 | \(-2, 4, -8, \ldots, -2048, 4096\) | -2 | 4096 | Finite |
| 5 | \(38, 33, 28, 23, 18, 13\) | 38 | 13 | Finite |

- Activity 3:
1. 110
2. 0.5
3. 91
4. 13
5. \(\frac{9}{14}\)

Final Boxed Answer:
\[
\boxed{
\begin{array}{l}
\text{Activity 1: SEQUENCE: A, D; SERIES: B, C} \\
\text{Activity 2: See table above} \\
\text{Activity 3: 110, 0.5, 91, 13, } \frac{9}{14}
\end{array}
}
\]