Let's solve the problem step by step:
(a) Finding the 19th term of the arithmetic series:
First, we will find the common difference ‘d’, and the first term ‘a’.
We are given two terms from the arithmetic series:
The 4th term (A4) is 33.
The 10th term (A10) is 69.
The nth term in an arithmetic series can be found with the formula:
An = A1 + (n-1)d
Where:
An is the nth term,
A1 is the first term, and
d is the common difference.
From the given information, we can set up two equations based on the formula above:
For A4: A1 + 3d = 33 (1)
For A10: A1 + 9d = 69 (2)
Now, we will find the value of ‘d’ (common difference) by subtracting equation (1) from equation (2):
(A1 + 9d) - (A1 + 3d) = 69 - 33
9d - 3d = 36
6d = 36
d = 36 / 6
d = 6
Now we have the common difference, let's find the first term ‘a’ (A1). We can use either equation (1) or (2) to find it. Let’s use equation (1):
A1 + 3d = 33
A1 + 3(6) = 33
A1 + 18 = 33
A1 = 33 - 18
A1 = 15
Now that we have the first term A1 and the common difference d, we can find the 19th term (A19) using the formula:
A19 = A1 + 18d
A19 = 15 + 18(6)
A19 = 15 + 108
A19 = 123
The 19th term of the arithmetic series is 123.
(b) The rule for the sequence:
For an arithmetic series, the rule or the nth term formula is:
An = A1 + (n-1)d
We have A1 = 15 (first term) and d = 6 (common difference), so the rule for the sequence is:
An = 15 + (n-1)6
(c) The common difference 'd' and first term 'a':
The common difference 'd' is 6.
The first term 'a' (A1) is 15.
Therefore, we have found:
The 19th term, which is 123.
The rule for the sequence, which is An = 15 + (n-1)6.
The common difference 'd', which is 6, and the first term 'a', which is 15.
