Find the mean and the median in the following arithmetic sequences:
A: 1, 3, 5, 7, 9
B: 1, 3, 5, 7, 9, . . . ., 199
C: 7, 10, 13, 16, . . ., 607
OK: I'll show you how to find n (how many numbers in the sequence) as well. You'll most likely need to now how.
A. (1+3+5+7+9)/5 is a mean of 5 and a median of 5.
- This one is simple enough, just add them and divide by how many there are.
And of course, look in the middle for the median.
B. (1+3+5+7+9...199) is a mean of
-A little more complicated, the series has a common difference: d of 2 and goes from 1 to 199. We don't know how many numbers are in this series though.
To solve for this:n, use the formula:
a1+d(n-1)=ax
a1+2(n-1)=ax
1+2(n-1)=199
1+2n-2=199
2n-2=198
2n=200
n=100
We now know that there are 100 numbers in the series.
Solving for mean and median:
(a1+ax)/2, simply add the first and last and divide by 2.
(1+199)/2 = 200/2 = 100 is the mean.
The median of an arithmetic sequence is just the average, so it is 100 as well.
C. (7,10,13,16...607)
-Same idea as B.
The common difference: d is 3.
Finding n....
a1+d(n-1)=ax
a1+3(n-1)=ax
7+3(n-1)=607
7+3n-3=607
3n-3=600
3n=603
n=201
Now to find the mean and median, which are the same in an arithmetic sequence:
(a1+ax)/2 = (7+607)/2 = 614/2 =307 is the mean and median.
SO:
A. Mean and Median: 5
B. Mean and Median: 100
C. Mean and Median: 307
Adding the first and last and dividing by 2 works because each number is an equal distance apart.
To find the mean and median of an arithmetic sequence, we need to understand the properties of the sequence.
- An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.
Let's calculate the mean and median for each of the given arithmetic sequences:
A: 1, 3, 5, 7, 9
Mean:
To find the mean, we sum up all the terms in the sequence and then divide by the total number of terms.
Sum of the terms: 1 + 3 + 5 + 7 + 9 = 25
Total number of terms: 5
Mean = Sum of terms / Total number of terms
Mean = 25 / 5
Mean = 5
So, the mean of sequence A is 5.
Median:
To find the median, we need to arrange the terms in the sequence in ascending order.
Arranged sequence: 1, 3, 5, 7, 9
As the total number of terms is odd, the median is the middle term.
The middle term in this case is the third term, which is 5.
So, the median of sequence A is 5.
B: 1, 3, 5, 7, 9, . . . ., 199
Mean:
As the sequence continues until 199, we can observe that the sequence is an arithmetic sequence with a common difference of 2.
To find the number of terms in this sequence, we can use the formula for the nth term of an arithmetic sequence:
nth term = first term + (n - 1) * common difference
199 = 1 + (n - 1) * 2
198 = (n - 1) * 2
99 = (n - 1)
n = 100
So, there are 100 terms in the sequence.
Sum of the terms: (1 + 199) / 2 * 100 = 10,000
Total number of terms: 100
Mean = Sum of terms / Total number of terms
Mean = 10,000 / 100
Mean = 100
So, the mean of sequence B is 100.
Median:
To find the median, we need to find the middle term of the sequence.
As there are 100 terms in the sequence, the middle term is the 50th term.
nth term = first term + (n - 1) * common difference
50th term = 1 + (50 - 1) * 2
50th term = 1 + 98
50th term = 99
So, the median of sequence B is 99.
C: 7, 10, 13, 16, . . ., 607
Mean:
As the sequence continues until 607, we can observe that the sequence is an arithmetic sequence with a common difference of 3.
To find the number of terms in this sequence, we can use the formula for the nth term of an arithmetic sequence:
nth term = first term + (n - 1) * common difference
607 = 7 + (n - 1) * 3
600 = (n - 1) * 3
200 = n - 1
n = 201
So, there are 201 terms in the sequence.
Sum of the terms: (7 + 607) / 2 * 201 = 61,212
Total number of terms: 201
Mean = Sum of terms / Total number of terms
Mean = 61,212 / 201
Mean ≈ 304.39
So, the mean of sequence C is approximately 304.39.
Median:
To find the median, we need to find the middle term of the sequence.
As there are 201 terms in the sequence, the middle term is the 101st term.
nth term = first term + (n - 1) * common difference
101st term = 7 + (101 - 1) * 3
101st term = 7 + 300
101st term = 307
So, the median of sequence C is 307.