In an A.P the difference between the 8th term and 4th term is 20 and the 8th term is 1half timesthe fourth term. what is the common difference and the first term of the sequence
T8-T4 = 4d, so d=5
T8 = T4/2
(a+7d) = (a+3d)/2
a+35 = (a+15)/2
2a+70 = a+15
a = -55
To find the common difference and the first term of the arithmetic progression (A.P), we can use the given information about the difference between the terms.
Let's denote the common difference as 'd' and the first term as 'a'.
1. The difference between the 8th term and the 4th term is given as 20.
So, the 8th term (denoted as A[8]) minus the 4th term (denoted as A[4]) equals 20.
A[8] - A[4] = 20
Since the common difference is constant for an arithmetic progression, the difference between any two terms should be equal to 4 times the common difference.
A[8] - A[4] = 4d
Replacing the expression for the difference:
4d = 20
Dividing both sides by 4:
d = 5
So, the common difference 'd' is 5.
2. The 8th term is stated to be one-half times the 4th term.
Mathematically, this can be represented as:
A[8] = (1/2) * A[4]
Substituting the values of the 8th and 4th terms:
(1/2) * A[4] = A[8]
Since the nth term of an A.P can be represented as: A[n] = a + (n-1)d
We can use this formula to express the 4th and 8th terms in terms of 'a' and 'd'.
A[4] = a + 3d (since n-1 = 4-1 = 3)
A[8] = a + 7d (since n-1 = 8-1 = 7)
Substituting these expressions:
(1/2)(a + 3d) = a + 7d
Simplifying the equation:
(1/2)a + (3/2)d = a + 7d
(1/2)a = (7/2)d
Dividing both sides by (1/2):
a = (7/2)d
Substituting the value of 'd' as 5:
a = (7/2) * 5
a = 35/2
a = 17.5
Therefore, the first term of the arithmetic progression is 17.5.
To summarize:
The common difference (d) is 5.
The first term (a) is 17.5.