The fourth term of an Ap is 9 and the sum of the first ten terms is 60. Find the first term and the common difference?
a + 3d = 9
10/2 (2a + 9d) = 60
Now just solve for a and d.
How will we solve that problem
To find the first term and the common difference of an arithmetic progression (AP), we can use the given information.
Let's denote the first term as 'a' and the common difference as 'd'.
We are given that the fourth term of the AP is 9, which means a + 3d = 9. This is because the fourth term is obtained by adding the common difference three times to the first term.
We are also given that the sum of the first ten terms is 60. The sum of an arithmetic progression can be found using the formula Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the first 'n' terms.
Using this formula, we can set up the equation:
60 = (10/2)(2a + (10-1)d)
60 = 5(2a + 9d)
12 = 2a + 9d
Now we have a system of equations:
a + 3d = 9 (equation 1)
2a + 9d = 12 (equation 2)
We can solve this system of equations to find the values of 'a' and 'd'.
By solving equations 1 and 2, we can eliminate 'a'.
Multiply equation 1 by 2:
2a + 6d = 18 (equation 3)
Subtract equation 3 from equation 2:
(2a + 9d) - (2a + 6d) = 12 - 18
3d = -6
d = -2
Now substitute the value of 'd' into equation 1 to find 'a':
a + 3(-2) = 9
a - 6 = 9
a = 9 + 6
a = 15
Therefore, the first term of the arithmetic progression is 15, and the common difference is -2.