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**Republic of the Philippines**
**Department of Education**
**REGION VI - WESTERN VISAYAS**
**SCHOOLS DIVISION OF ESCALANTE CITY**
**ESCALANTE NATIONAL HIGH SCHOOL**
**FIRST QUARTER EXAMINATION IN MATHEMATICS 10**
**SCHOOL YEAR 2024 - 2025**

**DIRECTIONS:** Read each item carefully and comprehensively. For selected-response items, WRITE the LETTER that corresponds to the correct answer. For constructed-response items, SOLVE for the answer. Show the process and the solution in your answer sheet.

1. Which among the numerical patterns below generates the pattern "the next term is obtained by alternately subtracting 5 and multiplying by [tex]$2$[/tex]," when starting with [tex]$-3$[/tex]?
A. [tex]$-3, 8, -16, 21, \ldots$[/tex]
B. [tex]$-3, 8, 16, -21, \ldots$[/tex]
C. [tex]$-3, -8, -16, 21, \ldots$[/tex]
D. [tex]$-3, -8, -16, -21, \ldots$[/tex]

2. Daniel has the number pattern [tex]$\{5, 10, 8, 13, 11, \ldots\}$[/tex] for a homework problem. How did Daniel generate the pattern?

3. Which of the following patterns below illustrates an arithmetic sequence?
A. [tex]$3, 6, 12, 24, \ldots$[/tex]
B. [tex]$4.1, 11.1, 18.1, 25.1, \ldots$[/tex]
C. [tex]$-2, 6, -18, 54, \ldots$[/tex]
D. [tex]$36, 18, 9, 4.5, \ldots$[/tex]

4. Illustrate the first 5 terms of an arithmetic sequence whose first term is [tex]$3$[/tex] and a common difference of [tex]$-2$[/tex]. Write your answer below:

5. Determine the common difference in the sequence: [tex]$-7, -3, 1, 5, 9, \ldots$[/tex]
A. [tex]$-3$[/tex]
B. [tex]$3$[/tex]
C. [tex]$4$[/tex]
D. [tex]$-4$[/tex]

6. Determine the next four terms generated in the sequence: [tex]$3, 9, 15, 21, \ldots$[/tex]
A. [tex]$27, 33, 39, 44$[/tex]
B. [tex]$27, 33, 39, 45$[/tex]
C. [tex]$27, 33, 39, 46$[/tex]
D. [tex]$27, 33, 39, 47$[/tex]

7. Determine the [tex]$8^{\text{th}}$[/tex] term in the sequence [tex]$-1, 4, 9, 14, 19, \ldots$[/tex]
A. [tex]$34$[/tex]
B. [tex]$35$[/tex]
C. [tex]$36$[/tex]
D. [tex]$37$[/tex]

Answer :

Sure, let's go through each question and answer it step-by-step.

### Question 1
Given:
Start with -3 and generate the pattern where the next term is obtained by alternately subtracting 5 and multiplying by 2.

First term: -3
1. Subtract 5: -3 - 5 = -8
2. Multiply by 2: -8 * 2 = -16
3. Subtract 5: -16 - 5 = -21

Now, we find the answer choice:
- Option A: -3, 8, -16, 21, ...
- Option B: -3, 8, 16, -21, ...
- Option C: -3, -8, -16, 21, ...
- Option D: -3, -8, -16, -21, ...

Answer: D

### Question 2
Pattern given: {5, 10, 8, 13, 11, ...}

By observing the pattern:
1. 5 + 5 = 10
2. 10 - 2 = 8
3. 8 + 5 = 13
4. 13 - 2 = 11

So, the pattern alternately adds 5 and then subtracts 2.

Answer: The pattern alternately adds 5 and then subtracts 2.

### Question 3
Given options to identify an arithmetic sequence (common difference between the terms):
- Option A: 3, 6, 12, 24, ... (Geometric sequence)
- Option B: 4.1, 11.1, 18.1, 25.1, ... (Common difference = 7)
- Option C: -2, 6, -18, 54, ... (Neither arithmetic nor geometric)
- Option D: 36, 18, 9, 4.5, ... (Geometric sequence)

Answer: B

### Question 4
First term: 3 \
Common difference: -2

1. First term: 3
2. Second term: 3 + (-2) = 1
3. Third term: 1 + (-2) = -1
4. Fourth term: -1 + (-2) = -3
5. Fifth term: -3 + (-2) = -5

Answer: 3, 1, -1, -3, -5

### Question 5
Given sequence: -7, -3, 1, 5, 9

We need to determine the common difference:
- Difference between -3 and -7: -3 - (-7) = 4
- Difference between 1 and -3: 1 - (-3) = 4
- Difference between 5 and 1: 5 - 1 = 4
- Difference between 9 and 5: 9 - 5 = 4

Answer: C (Common difference is 4)

### Question 6
Given initial sequence: 3, 9, 15, 21

Identifying the common difference:
- Difference between 9 and 3: 6
- Difference between 15 and 9: 6
- Difference between 21 and 15: 6

Next terms:
1. 21 + 6 = 27
2. 27 + 6 = 33
3. 33 + 6 = 39
4. 39 + 6 = 45

Answer: B (Next terms are 27, 33, 39, 45)

### Question 7
Given sequence: -1, 4, 9, 14, 19

We observe the sequence has a common difference of 5.

To find the 8th term:
1. First term: -1
2. Common difference: 5
3. nth term formula for an arithmetic sequence: [tex]\(a_n = a_1 + (n-1) \cdot d\)[/tex]

[tex]\[
a_8 = -1 + (8-1) \cdot 5 = -1 + 35 = 34
\][/tex]

Answer: A

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Rewritten by : Barada