Answer :
Of course! Let's determine the number of terms in each arithmetic progression step-by-step.
### Sequence a: [tex]\((-3, -2, -1, 0, 1, 2, 3)\)[/tex]
1. Identify the first term ([tex]\(a_1\)[/tex]) and the last term:
- First term ([tex]\(a_1\)[/tex]) = [tex]\(-3\)[/tex]
- Last term = 3
2. Calculate the number of terms:
- The common difference ([tex]\(d\)[/tex]) here is 1 (since each term increases by 1).
- Number of terms, [tex]\(n\)[/tex], is given by:
[tex]\[
n = \text{(Last term - First term) / Common difference} + 1 = (3 - (-3)) / 1 + 1 = 7
\][/tex]
### Sequence b: [tex]\((120, 115, 110, 105, 100)\)[/tex]
1. Identify the first term and the last term:
- First term = 120
- Last term = 100
2. Calculate the number of terms:
- The common difference ([tex]\(d\)[/tex]) is [tex]\(-5\)[/tex] (since each term decreases by 5).
- Number of terms, [tex]\(n\)[/tex]:
[tex]\[
n = \frac{120 - 100}{5} + 1 = 5
\][/tex]
### Sequence c: [tex]\((3, 7, 11, \ldots, 23)\)[/tex]
1. Identify the first term and the last term:
- First term = 3
- Last term = 23
2. Calculate the number of terms:
- The common difference ([tex]\(d\)[/tex]) is 4.
- Number of terms, [tex]\(n\)[/tex]:
[tex]\[
n = \frac{23 - 3}{4} + 1 = 6
\][/tex]
### Sequence d: [tex]\((5, 8, 11, \ldots, 20)\)[/tex]
1. Identify the first term and the last term:
- First term = 5
- Last term = 20
2. Calculate the number of terms:
- The common difference ([tex]\(d\)[/tex]) is 3.
- Number of terms, [tex]\(n\)[/tex]:
[tex]\[
n = \frac{20 - 5}{3} + 1 = 6
\][/tex]
By following these steps, we can determine that the number of terms in these sequences are 7, 5, 6, and 6 respectively.
### Sequence a: [tex]\((-3, -2, -1, 0, 1, 2, 3)\)[/tex]
1. Identify the first term ([tex]\(a_1\)[/tex]) and the last term:
- First term ([tex]\(a_1\)[/tex]) = [tex]\(-3\)[/tex]
- Last term = 3
2. Calculate the number of terms:
- The common difference ([tex]\(d\)[/tex]) here is 1 (since each term increases by 1).
- Number of terms, [tex]\(n\)[/tex], is given by:
[tex]\[
n = \text{(Last term - First term) / Common difference} + 1 = (3 - (-3)) / 1 + 1 = 7
\][/tex]
### Sequence b: [tex]\((120, 115, 110, 105, 100)\)[/tex]
1. Identify the first term and the last term:
- First term = 120
- Last term = 100
2. Calculate the number of terms:
- The common difference ([tex]\(d\)[/tex]) is [tex]\(-5\)[/tex] (since each term decreases by 5).
- Number of terms, [tex]\(n\)[/tex]:
[tex]\[
n = \frac{120 - 100}{5} + 1 = 5
\][/tex]
### Sequence c: [tex]\((3, 7, 11, \ldots, 23)\)[/tex]
1. Identify the first term and the last term:
- First term = 3
- Last term = 23
2. Calculate the number of terms:
- The common difference ([tex]\(d\)[/tex]) is 4.
- Number of terms, [tex]\(n\)[/tex]:
[tex]\[
n = \frac{23 - 3}{4} + 1 = 6
\][/tex]
### Sequence d: [tex]\((5, 8, 11, \ldots, 20)\)[/tex]
1. Identify the first term and the last term:
- First term = 5
- Last term = 20
2. Calculate the number of terms:
- The common difference ([tex]\(d\)[/tex]) is 3.
- Number of terms, [tex]\(n\)[/tex]:
[tex]\[
n = \frac{20 - 5}{3} + 1 = 6
\][/tex]
By following these steps, we can determine that the number of terms in these sequences are 7, 5, 6, and 6 respectively.
