Answer :
Sure, I'll provide detailed step-by-step solutions for each question given the correct answers.
### Question 1
In an arithmetic progression, [tex]\(a_1 = 15\)[/tex] and [tex]\(d = 3\)[/tex]. Find the sum up to 5 terms.
Correct Answer: B. 105
Explanation:
The sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of an arithmetic progression is given by:
[tex]\[ S_n = \frac{n}{2} \left( 2a_1 + (n-1)d \right) \][/tex]
Given:
- [tex]\(a_1 = 15\)[/tex]
- [tex]\(d = 3\)[/tex]
- [tex]\(n = 5\)[/tex]
Plug in these values:
[tex]\[ S_5 = \frac{5}{2} \left( 2 \cdot 15 + (5-1) \cdot 3 \right) \][/tex]
[tex]\[ S_5 = \frac{5}{2} \left( 30 + 12 \right) \][/tex]
[tex]\[ S_5 = \frac{5}{2} \cdot 42 \][/tex]
[tex]\[ S_5 = 5 \cdot 21 \][/tex]
[tex]\[ S_5 = 105 \][/tex]
So, the sum of the first 5 terms is 105.
### Question 2
Which of the following is an arithmetic series?
- A. [tex]\(3, 8, 11, 19, 30\)[/tex]
- C. [tex]\(2, 7, 12, 18, 24\)[/tex]
- B. [tex]\(2, 4, 8, 16, 32\)[/tex]
- D. [tex]\(12, 7, 2, -3, -8\)[/tex]
Correct Answer: D
Explanation:
An arithmetic series has a constant difference between consecutive terms. Let's check each series:
- Series A: [tex]\(3, 8, 11, 19, 30\)[/tex]
- Differences: [tex]\(8-3 = 5\)[/tex], [tex]\(11-8 = 3\)[/tex], [tex]\(19-11 = 8\)[/tex], [tex]\(30-19 = 11\)[/tex]
- The differences are not constant.
- Series C: [tex]\(2, 7, 12, 18, 24\)[/tex]
- Differences: [tex]\(7-2 = 5\)[/tex], [tex]\(12-7 = 5\)[/tex], [tex]\(18-12 = 6\)[/tex], [tex]\(24-18 = 6\)[/tex]
- The differences are not constant.
- Series B: [tex]\(2, 4, 8, 16, 32\)[/tex]
- Differences: [tex]\(4-2 = 2\)[/tex], [tex]\(8-4 = 4\)[/tex], [tex]\(16-8 = 8\)[/tex], [tex]\(32-16 = 16\)[/tex]
- The differences are not constant (actually a geometric series).
- Series D: [tex]\(12, 7, 2, -3, -8\)[/tex]
- Differences: [tex]\(7-12 = -5\)[/tex], [tex]\(2-7 = -5\)[/tex], [tex]\(-3-2 = -5\)[/tex], [tex]\(-8-(-3) = -5\)[/tex]
- The difference is constant.
Hence, the correct arithmetic series is D. [tex]\(12, 7, 2, -3, -8\)[/tex].
### Question 3
What is the sum of all even numbers between 1 and 35?
- A. 300
- B. 306
- C. 308
- D. 310
Correct Answer: B. 306
Explanation:
The even numbers between 1 and 35 are: [tex]\(2, 4, 6, ..., 34\)[/tex].
This is an arithmetic sequence with:
- First term [tex]\(a_1 = 2\)[/tex]
- Common difference [tex]\(d = 2\)[/tex]
- Last term [tex]\(a_n = 34\)[/tex]
Using the formula for the sum of an arithmetic series [tex]\(S_n = \frac{n}{2} (a_1 + a_n)\)[/tex], we need to find the number of terms [tex]\(n\)[/tex]:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
[tex]\[ 34 = 2 + (n-1) \cdot 2 \][/tex]
[tex]\[ 34 = 2 + 2n - 2 \][/tex]
[tex]\[ 34 = 2n \][/tex]
[tex]\[ n = 17 \][/tex]
Now calculate the sum:
[tex]\[ S_{17} = \frac{17}{2} (2 + 34) \][/tex]
[tex]\[ S_{17} = \frac{17}{2} \cdot 36 \][/tex]
[tex]\[ S_{17} = 17 \cdot 18 \][/tex]
[tex]\[ S_{17} = 306 \][/tex]
Therefore, the sum of all even numbers between 1 and 35 is 306.
### Question 4
Which of the following is the sum of the first 8 multiples of 5?
Correct Answer: 180
Explanation:
The first 8 multiples of 5 are [tex]\(5, 10, 15, 20, 25, 30, 35, 40\)[/tex].
Using the formula for the sum of an arithmetic series:
[tex]\[ \text{Sum} = 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 \][/tex]
This is an arithmetic sequence with:
- First term [tex]\(a_1 = 5\)[/tex]
- Common difference [tex]\(d = 5\)[/tex]
- Number of terms [tex]\(n = 8\)[/tex]
- Last term [tex]\(a_8 = 40\)[/tex]
The sum can be found using the sum formula for an arithmetic series:
[tex]\[ S_n = \frac{n}{2} (a_1 + a_n) \][/tex]
[tex]\[ S_8 = \frac{8}{2} (5 + 40) \][/tex]
[tex]\[ S_8 = 4 \cdot 45 \][/tex]
[tex]\[ S_8 = 180 \][/tex]
Therefore, the sum of the first 8 multiples of 5 is 180.
### Question 1
In an arithmetic progression, [tex]\(a_1 = 15\)[/tex] and [tex]\(d = 3\)[/tex]. Find the sum up to 5 terms.
Correct Answer: B. 105
Explanation:
The sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of an arithmetic progression is given by:
[tex]\[ S_n = \frac{n}{2} \left( 2a_1 + (n-1)d \right) \][/tex]
Given:
- [tex]\(a_1 = 15\)[/tex]
- [tex]\(d = 3\)[/tex]
- [tex]\(n = 5\)[/tex]
Plug in these values:
[tex]\[ S_5 = \frac{5}{2} \left( 2 \cdot 15 + (5-1) \cdot 3 \right) \][/tex]
[tex]\[ S_5 = \frac{5}{2} \left( 30 + 12 \right) \][/tex]
[tex]\[ S_5 = \frac{5}{2} \cdot 42 \][/tex]
[tex]\[ S_5 = 5 \cdot 21 \][/tex]
[tex]\[ S_5 = 105 \][/tex]
So, the sum of the first 5 terms is 105.
### Question 2
Which of the following is an arithmetic series?
- A. [tex]\(3, 8, 11, 19, 30\)[/tex]
- C. [tex]\(2, 7, 12, 18, 24\)[/tex]
- B. [tex]\(2, 4, 8, 16, 32\)[/tex]
- D. [tex]\(12, 7, 2, -3, -8\)[/tex]
Correct Answer: D
Explanation:
An arithmetic series has a constant difference between consecutive terms. Let's check each series:
- Series A: [tex]\(3, 8, 11, 19, 30\)[/tex]
- Differences: [tex]\(8-3 = 5\)[/tex], [tex]\(11-8 = 3\)[/tex], [tex]\(19-11 = 8\)[/tex], [tex]\(30-19 = 11\)[/tex]
- The differences are not constant.
- Series C: [tex]\(2, 7, 12, 18, 24\)[/tex]
- Differences: [tex]\(7-2 = 5\)[/tex], [tex]\(12-7 = 5\)[/tex], [tex]\(18-12 = 6\)[/tex], [tex]\(24-18 = 6\)[/tex]
- The differences are not constant.
- Series B: [tex]\(2, 4, 8, 16, 32\)[/tex]
- Differences: [tex]\(4-2 = 2\)[/tex], [tex]\(8-4 = 4\)[/tex], [tex]\(16-8 = 8\)[/tex], [tex]\(32-16 = 16\)[/tex]
- The differences are not constant (actually a geometric series).
- Series D: [tex]\(12, 7, 2, -3, -8\)[/tex]
- Differences: [tex]\(7-12 = -5\)[/tex], [tex]\(2-7 = -5\)[/tex], [tex]\(-3-2 = -5\)[/tex], [tex]\(-8-(-3) = -5\)[/tex]
- The difference is constant.
Hence, the correct arithmetic series is D. [tex]\(12, 7, 2, -3, -8\)[/tex].
### Question 3
What is the sum of all even numbers between 1 and 35?
- A. 300
- B. 306
- C. 308
- D. 310
Correct Answer: B. 306
Explanation:
The even numbers between 1 and 35 are: [tex]\(2, 4, 6, ..., 34\)[/tex].
This is an arithmetic sequence with:
- First term [tex]\(a_1 = 2\)[/tex]
- Common difference [tex]\(d = 2\)[/tex]
- Last term [tex]\(a_n = 34\)[/tex]
Using the formula for the sum of an arithmetic series [tex]\(S_n = \frac{n}{2} (a_1 + a_n)\)[/tex], we need to find the number of terms [tex]\(n\)[/tex]:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
[tex]\[ 34 = 2 + (n-1) \cdot 2 \][/tex]
[tex]\[ 34 = 2 + 2n - 2 \][/tex]
[tex]\[ 34 = 2n \][/tex]
[tex]\[ n = 17 \][/tex]
Now calculate the sum:
[tex]\[ S_{17} = \frac{17}{2} (2 + 34) \][/tex]
[tex]\[ S_{17} = \frac{17}{2} \cdot 36 \][/tex]
[tex]\[ S_{17} = 17 \cdot 18 \][/tex]
[tex]\[ S_{17} = 306 \][/tex]
Therefore, the sum of all even numbers between 1 and 35 is 306.
### Question 4
Which of the following is the sum of the first 8 multiples of 5?
Correct Answer: 180
Explanation:
The first 8 multiples of 5 are [tex]\(5, 10, 15, 20, 25, 30, 35, 40\)[/tex].
Using the formula for the sum of an arithmetic series:
[tex]\[ \text{Sum} = 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 \][/tex]
This is an arithmetic sequence with:
- First term [tex]\(a_1 = 5\)[/tex]
- Common difference [tex]\(d = 5\)[/tex]
- Number of terms [tex]\(n = 8\)[/tex]
- Last term [tex]\(a_8 = 40\)[/tex]
The sum can be found using the sum formula for an arithmetic series:
[tex]\[ S_n = \frac{n}{2} (a_1 + a_n) \][/tex]
[tex]\[ S_8 = \frac{8}{2} (5 + 40) \][/tex]
[tex]\[ S_8 = 4 \cdot 45 \][/tex]
[tex]\[ S_8 = 180 \][/tex]
Therefore, the sum of the first 8 multiples of 5 is 180.
