the 4th term of an A.p is 8,the sum of first 12 terms is 126.Find the A.p
a + 3 d = 8 ... 12 a + 36 d = 96
12 a + 66 d = 126
subtracting equations ... 30 d = 30
substitute back to find a
Help me solve it out
Just use the formulas you know, and what they have told you.
a+3d = 8
12/2 (2a+11d) = 126
Now just solve for a and d.
I don't understand pls
To find the arithmetic progression (AP), we need to find the common difference (d) and the first term (a).
Step 1: Find the common difference (d)
We are given that the 4th term of the AP is 8. This means that the 4th term can be represented as a + 3d (since the difference between each term is d).
Therefore, a + 3d = 8 ------(1)
Step 2: Find the first term (a)
We are given that the sum of the first 12 terms is 126. The formula to find the sum of an AP is:
Sn = (n / 2) * [2a + (n - 1) * d]
Substituting the given values:
126 = (12 / 2) * [2a + (12 - 1) * d]
126 = 6 * [2a + 11d]
126 = 12a + 66d ------(2)
Step 3: Solve the equations (1) and (2) simultaneously to find the values of a and d.
We can solve the equations (1) and (2) to find the values of a and d using any method such as substitution or elimination.
Let's use the elimination method.
Multiply equation (1) by 6:
6a + 18d = 48 ------(3)
Subtract equation (3) from equation (2):
(12a + 66d) - (6a + 18d) = 126 - 48
6a + 48d = 78
Divide the equation by 6:
a + 8d = 13 ------(4)
Now we have two equations:
a + 8d = 13 ------(4)
a + 3d = 8 ------(1)
Subtract equation (1) from equation (4):
(a + 8d) - (a + 3d) = 13 - 8
8d - 3d = 5d = 5
Divide the equation by 5:
d = 1
Substitute the value of d = 1 into equation (1):
a + 3(1) = 8
a + 3 = 8
a = 8 - 3
a = 5
Step 4: Write the arithmetic progression
From the values of a and d, we can now write the AP as follows:
a, a + d, a + 2d, a + 3d, ...
Substituting the values of a = 5 and d = 1:
5, 5 + 1, 5 + 2, 5 + 3, ...
5, 6, 7, 8, ...
Therefore, the given AP is: 5, 6, 7, 8, ...