The fourth term of an AP sequence is 1less than twice the second term. If the sixth term is 7.Find the first term.
Answer
so you have
a+3d = 2(a+2d)-1
a+5d = 7
Now just solve for a
To find the first term of an arithmetic progression (AP) sequence, we need to know the common difference and at least one term of the sequence.
Let's represent the first term of the sequence as 'a', and the common difference as 'd'.
We are given that the fourth term of the sequence is 1 less than twice the second term. So we can express this relationship as:
a + 3d = 2(a + d) - 1
Now, we are also given that the sixth term of the sequence is 7. So we can express this relationship as:
a + 5d = 7
We have a system of two equations with two variables. Let's solve this system of equations to find the values of 'a' and 'd'.
First, simplify the equation a + 3d = 2(a + d) - 1:
a + 3d = 2a + 2d - 1
Rearrange the terms:
a - 2a + 3d - 2d = -1
Simplify further:
-a + d = -1
Now, we have two equations:
-a + d = -1 (equation 1)
a + 5d = 7 (equation 2)
To solve this system, we can eliminate 'a' by adding equation 1 and equation 2:
(-a + d) + (a + 5d) = -1 + 7
Simplify:
6d = 6
Divide both sides of the equation by 6:
d = 1
Now, substitute the value of 'd' into equation 1 to find 'a':
-a + 1 = -1
Add 'a' to both sides of the equation:
1 = a
Therefore, the first term of the AP sequence is 1.
So, the first term of the AP sequence is 1.