An arithmetic series has third term 11. The ninth term is 5 times the second term. Find the common difference.
Let the first term be called a and the common difference be called b.
a + 2b = 11
a + 8b = 5*(a + b) = 5a + 5b
The last equation can be rewritten
4a - 3b = 0
4a + 8b = 44
11b = 44
b = 4
a = 11 - 2b = 3
The series is 3,7,11,15,19,23,27,31,35...
Note that 35 (the ninth term) is 5 times 7
To find the common difference in an arithmetic series, we need to use the given information.
Let's write out the terms of the arithmetic series:
a, a + d, a + 2d, a + 3d, ...
We are given that the third term is 11, so we can say:
a + 2d = 11 ...(1)
We are also given that the ninth term is 5 times the second term, so we can say:
a + 8d = 5(a + d) ...(2)
Now we have a system of equations. We can solve this system to find the common difference.
First, let's simplify equation (2):
a + 8d = 5a + 5d
8d - 5d = 5a - a
3d = 4a ...(3)
Next, substitute equation (3) into equation (1):
a + 2(4a/3) = 11
a + 8a/3 = 11
(3a + 8a)/3 = 11
11a/3 = 11
11a = 33
a = 3
Now we can substitute the value of 'a' back into equation (3) to find the common difference:
3d = 4a
3d = 4(3)
3d = 12
d = 4
Therefore, the common difference in this arithmetic series is 4.