Consider the arithmetic series 7 + 19 + 31 + 43 + 55 + pr004-1.jpg.
a) How many terms are less than 1000? (3 marks - show your work)
b) How many terms are less than 2000? (3 marks - show your work)
We have
a=7
d=12
So, when does Tn exceed 1000?
Tn = 7+12(n-1) = 12n-5
12n-5 < 1000
12n < 1005
n < 83.75
So, the first 83 terms are less than 1000
Similarly for 2000
To find the number of terms in an arithmetic series, we can use the formula:
Number of terms = (Last term - First term) / Common difference + 1
Given that the first term is 7 and the common difference is 12 (the difference between consecutive terms), we can use this information to solve the questions.
a) To find the number of terms less than 1000:
First, we calculate the last term using the formula:
Last term = First term + (Number of terms - 1) * Common difference
Since the last term is not known, we can use the formula in reverse to find the number of terms:
Last term = First term + (Number of terms - 1) * Common difference
Substituting the known values:
1000 = 7 + (Number of terms - 1) * 12
Simplifying the equation:
993 = (Number of terms - 1) * 12
Dividing both sides by 12:
Number of terms - 1 = 993 / 12
Number of terms - 1 = 82.75
Adding 1 to both sides:
Number of terms = 82.75 + 1
Number of terms = 83.75
Since the number of terms must be a whole number, we round down to the nearest whole number:
Number of terms = 83
Therefore, there are 83 terms in the given arithmetic series that are less than 1000.
b) To find the number of terms less than 2000:
Using the same formula, we can calculate the number of terms in a similar way:
Last term = First term + (Number of terms - 1) * Common difference
2000 = 7 + (Number of terms - 1) * 12
1993 = (Number of terms - 1) * 12
Number of terms - 1 = 1993 / 12
Number of terms - 1 = 166.083
Number of terms = 166.083 + 1
Number of terms = 167.083
Rounded down to the nearest whole number:
Number of terms = 167
Therefore, there are 167 terms in the given arithmetic series that are less than 2000.
To solve both parts of this question, we need to determine the pattern of the arithmetic series first. The first term, a, is 7 and the common difference, d, between each term is 12.
a) To find out how many terms are less than 1000, we need to find the term number (n) where the value is less than or equal to 1000.
We can use the formula for the general term of an arithmetic series:
an = a + (n - 1)d
In this case, the value for an should be less than or equal to 1000. Therefore, we can set up the equation:
7 + (n - 1)12 ≤ 1000
Simplifying this inequality:
7 + 12n - 12 ≤ 1000
12n - 5 ≤ 1000
12n ≤ 1005
n ≤ 1005/12
Now, we can round down 1005/12 to the nearest whole number, which gives us:
n ≤ 83.75
Therefore, the number of terms less than 1000 is 83 terms.
b) To find out how many terms are less than 2000, we will use the same approach.
Setting up the equation:
7 + (n - 1)12 ≤ 2000
Simplifying:
12n - 5 ≤ 2000
12n ≤ 2005
n ≤ 2005/12
Rounding down 2005/12 to the nearest whole number:
n ≤ 167.08
Therefore, the number of terms less than 2000 is 167 terms.