The 5th term of an arithmetic progression is 82 and the 12th term is 103.Find
First term and common difference
Let's denote the first term as "a" and the common difference as "d".
We know that the 5th term is 82, so we can write the equation:
a + 4d = 82 ---(Equation 1)
We also know that the 12th term is 103, so we can write the equation:
a + 11d = 103 ---(Equation 2)
Now we have a system of equations with two variables. We can solve this system to find the values of "a" and "d".
From Equation 2, we can rearrange it to isolate "a":
a = 103 - 11d
Substituting this expression for "a" into Equation 1, we get:
(103 - 11d) + 4d = 82
103 - 11d + 4d = 82
103 - 7d = 82
-7d = 82 - 103
-7d = -21
Dividing both sides of the equation by -7, we get:
d = -21 / -7
d = 3
Now that we have the value of "d", we can substitute it back into Equation 1 to solve for "a":
a + 4(3) = 82
a + 12 = 82
a = 70
Therefore, the first term is 70 and the common difference is 3.