an ap consists of 21 terms.the sum of 3terms in the middle is 129 and of the last three is 237 find the AP.
To solve this problem, we can use a system of equations. Let's denote the first term of the arithmetic progression as 'a' and the common difference as 'd'.
Given that the sum of 3 terms in the middle is 129, we can express this as:
(a + 10d) + (a + 11d) + (a + 12d) = 129
Expanding and simplifying, we get:
3a + 33d = 129 --------- Equation 1
Similarly, given that the sum of the last three terms is 237, we can express this as:
(a + 18d) + (a + 19d) + (a + 20d) = 237
Expanding and simplifying, we get:
3a + 57d = 237 --------- Equation 2
Now, we have a system of equations with two equations and two unknowns. We can solve this system to find the values of 'a' and 'd'.
Subtracting Equation 1 from Equation 2, we eliminate the variable 'a':
(3a + 57d) - (3a + 33d) = 237 - 129
24d = 108
Dividing both sides by 24, we get:
d = 4.5
Substituting back the value of 'd' into Equation 1, we can solve for 'a':
3a + 33(4.5) = 129
3a + 148.5 = 129
3a = 129 - 148.5
3a = -19.5
Dividing both sides by 3, we get:
a = -6.5
Therefore, the first term (a) of the arithmetic progression is -6.5 and the common difference (d) is 4.5. The 21 terms of the arithmetic progression can be found by adding the common difference consecutively to the first term:
-6.5, -2, 2.5, 7, 11.5, 16, ...
Please note that the above answer assumes that the arithmetic progression is increasing. If it's decreasing, the common difference will be negative.