Answer :
Let's work out the nth terms for the sequences given in the problem:
a) Sequence: 9, 12, 15, 18, ...
1. Identify the common difference (d):
The difference between consecutive terms is [tex]\(12 - 9 = 3\)[/tex].
So, the common difference [tex]\(d = 3\)[/tex].
2. Write the nth term formula for an arithmetic sequence:
The general form of the nth term for an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n-1) \times d \][/tex]
where [tex]\(a_1\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference.
3. Substitute the values:
Here, the first term [tex]\(a_1 = 9\)[/tex] and the common difference [tex]\(d = 3\)[/tex].
So the nth term formula is:
[tex]\[ a_n = 9 + (n-1) \times 3 \][/tex]
Simplify the expression:
[tex]\[ a_n = 9 + 3n - 3 = 3n + 6 \][/tex]
Therefore, the nth term for the sequence 9, 12, 15, 18, ... is [tex]\(3n + 6\)[/tex].
b) Sequence: 10, 15, 20, 25, ...
1. Identify the common difference (d):
The difference between consecutive terms is [tex]\(15 - 10 = 5\)[/tex].
So, the common difference [tex]\(d = 5\)[/tex].
2. Write the nth term formula for an arithmetic sequence:
The general form of the nth term is again:
[tex]\[ a_n = a_1 + (n-1) \times d \][/tex]
where [tex]\(a_1\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference.
3. Substitute the values:
Here, the first term [tex]\(a_1 = 10\)[/tex] and the common difference [tex]\(d = 5\)[/tex].
So the nth term formula is:
[tex]\[ a_n = 10 + (n-1) \times 5 \][/tex]
Simplify the expression:
[tex]\[ a_n = 10 + 5n - 5 = 5n + 5 \][/tex]
Therefore, the nth term for the sequence 10, 15, 20, 25, ... is [tex]\(5n + 5\)[/tex].
I hope this helps! Let me know if you have any more questions.
a) Sequence: 9, 12, 15, 18, ...
1. Identify the common difference (d):
The difference between consecutive terms is [tex]\(12 - 9 = 3\)[/tex].
So, the common difference [tex]\(d = 3\)[/tex].
2. Write the nth term formula for an arithmetic sequence:
The general form of the nth term for an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n-1) \times d \][/tex]
where [tex]\(a_1\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference.
3. Substitute the values:
Here, the first term [tex]\(a_1 = 9\)[/tex] and the common difference [tex]\(d = 3\)[/tex].
So the nth term formula is:
[tex]\[ a_n = 9 + (n-1) \times 3 \][/tex]
Simplify the expression:
[tex]\[ a_n = 9 + 3n - 3 = 3n + 6 \][/tex]
Therefore, the nth term for the sequence 9, 12, 15, 18, ... is [tex]\(3n + 6\)[/tex].
b) Sequence: 10, 15, 20, 25, ...
1. Identify the common difference (d):
The difference between consecutive terms is [tex]\(15 - 10 = 5\)[/tex].
So, the common difference [tex]\(d = 5\)[/tex].
2. Write the nth term formula for an arithmetic sequence:
The general form of the nth term is again:
[tex]\[ a_n = a_1 + (n-1) \times d \][/tex]
where [tex]\(a_1\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference.
3. Substitute the values:
Here, the first term [tex]\(a_1 = 10\)[/tex] and the common difference [tex]\(d = 5\)[/tex].
So the nth term formula is:
[tex]\[ a_n = 10 + (n-1) \times 5 \][/tex]
Simplify the expression:
[tex]\[ a_n = 10 + 5n - 5 = 5n + 5 \][/tex]
Therefore, the nth term for the sequence 10, 15, 20, 25, ... is [tex]\(5n + 5\)[/tex].
I hope this helps! Let me know if you have any more questions.
