The fourth term of A P is 37 and the 6th term is 12 more than the fourth term.find the first and the seven terms
Wonderful
clearly d=6 since T6 = T4+12
So,
T1 = T4-18 = 19
T7 = T4+18 = 55
What is the answer
To find the first term and the seventh term of an arithmetic progression (AP), we need first to determine the common difference.
Let's denote the first term as 'a' and the common difference as 'd'.
The formula to find the nth term of an AP is given by:
nth term (Tn) = a + (n - 1)d
We are given the following information:
The fourth term (T4) = 37
T4 = a + (4 - 1)d
37 = a + 3d -- Equation (1)
The sixth term (T6) is 12 more than the fourth term:
T6 = T4 + 12
T6 = a + (6 - 1)d
T6 = a + 5d
37 + 12 = a + 5d -- Equation (2)
By solving Equations (1) and (2) simultaneously, we can find the values of 'a' and 'd'.
Subtracting Equation (1) from Equation (2):
(37 + 12) - (a + 5d) = (a + 5d) - (a + 3d)
49 - a = 2d
From Equation (1):
37 = a + 3d
We now have a system of equations:
49 - a = 2d -- Equation (3)
37 = a + 3d -- Equation (4)
To solve this system, we can use the method of substitution.
Rearranging Equation (4):
a = 37 - 3d
Substituting this into Equation (3):
49 - (37 - 3d) = 2d
49 - 37 + 3d = 2d
12 + 3d - 2d = 0
d = -12
Substituting the value of 'd' back into Equation (4):
37 = a + 3(-12)
37 = a - 36
a = 73
Therefore, the first term is 73, and the common difference is -12.
To find the seventh term (T7), we can substitute the values of 'a' and 'd' into the formula for the nth term of an AP:
T7 = a + (7 - 1)d
T7 = 73 + (7 - 1)(-12)
T7 = 73 + 6(-12)
T7 = 73 - 72
T7 = 1
Hence, the first term is 73, and the seventh term is 1.