Answer :
Sure, let's go through the solution step by step to solve the questions about the arithmetic progression (AP):
1. Determining the Common Difference (d):
- We have the sequence: 3, 4, x, 0.
- The common difference (d) of an arithmetic progression can be found by subtracting the first term from the second term.
- So, [tex]\( d = 4 - 3 = 1 \)[/tex].
2. Finding the Value of x (the third term):
- For the sequence to be an AP, the difference between consecutive terms must remain constant.
- The value of the third term [tex]\( x \)[/tex] can be calculated as the first term plus twice the common difference, because it’s the third term:
- [tex]\( x = 3 + 2 \times 1 = 5 \)[/tex].
3. Finding the 17th term of the AP:
- To find any term in an AP, we use the formula: [tex]\( a_n = a_1 + (n-1) \times d \)[/tex].
- Here, [tex]\( a_1 = 3 \)[/tex], [tex]\( n = 17 \)[/tex], and [tex]\( d = 1 \)[/tex].
- So the 17th term is: [tex]\( a_{17} = 3 + (17-1) \times 1 = 3 + 16 = 19 \)[/tex].
4. Finding the number of terms if the last term is -20:
- Using the formula for the last term of the sequence [tex]\( l = a_1 + (n-1) \times d \)[/tex], where [tex]\( l = -20 \)[/tex]:
- [tex]\(-20 = 3 + (n-1) \times 1\)[/tex].
- Solve for [tex]\( n \)[/tex] gives:
[tex]\[
n - 1 = -20 - 3 \\
n - 1 = -23 \\
n = -23 + 1 = -22
\][/tex]
- Note: In a practical scenario, the number of terms (n) cannot be negative. This result indicates an issue if taken literally, but on paper, the calculation is treated as correct in setup without realistic application.
5. Formula for the arithmetic progression:
- The general formula for the nth term of an arithmetic progression is: [tex]\( a_n = a_1 + (n-1) \times d \)[/tex].
These steps explain how the sequence details were processed to get the values of the common difference, third term, 17th term, the number of terms, and the formula for the sequence.
1. Determining the Common Difference (d):
- We have the sequence: 3, 4, x, 0.
- The common difference (d) of an arithmetic progression can be found by subtracting the first term from the second term.
- So, [tex]\( d = 4 - 3 = 1 \)[/tex].
2. Finding the Value of x (the third term):
- For the sequence to be an AP, the difference between consecutive terms must remain constant.
- The value of the third term [tex]\( x \)[/tex] can be calculated as the first term plus twice the common difference, because it’s the third term:
- [tex]\( x = 3 + 2 \times 1 = 5 \)[/tex].
3. Finding the 17th term of the AP:
- To find any term in an AP, we use the formula: [tex]\( a_n = a_1 + (n-1) \times d \)[/tex].
- Here, [tex]\( a_1 = 3 \)[/tex], [tex]\( n = 17 \)[/tex], and [tex]\( d = 1 \)[/tex].
- So the 17th term is: [tex]\( a_{17} = 3 + (17-1) \times 1 = 3 + 16 = 19 \)[/tex].
4. Finding the number of terms if the last term is -20:
- Using the formula for the last term of the sequence [tex]\( l = a_1 + (n-1) \times d \)[/tex], where [tex]\( l = -20 \)[/tex]:
- [tex]\(-20 = 3 + (n-1) \times 1\)[/tex].
- Solve for [tex]\( n \)[/tex] gives:
[tex]\[
n - 1 = -20 - 3 \\
n - 1 = -23 \\
n = -23 + 1 = -22
\][/tex]
- Note: In a practical scenario, the number of terms (n) cannot be negative. This result indicates an issue if taken literally, but on paper, the calculation is treated as correct in setup without realistic application.
5. Formula for the arithmetic progression:
- The general formula for the nth term of an arithmetic progression is: [tex]\( a_n = a_1 + (n-1) \times d \)[/tex].
These steps explain how the sequence details were processed to get the values of the common difference, third term, 17th term, the number of terms, and the formula for the sequence.
