The 4th term of an a.p is 60. If the sum of 8th and 9th terms is -72, the common difference is
Let's denote the first term of the arithmetic progression as 'a' and the common difference as 'd'.
From the problem, we are given the following information:
The 4th term is 60:
a + 3d = 60 ------(1)
The sum of the 8th and 9th terms is -72:
(a + 7d) + (a + 8d) = -72
2a + 15d = -72 ------(2)
Now, we can solve these two equations simultaneously to find the values of 'a' and 'd'.
Let's multiply equation (1) by 2 and subtract equation (2) from it:
2a + 6d - (2a + 15d) = 120 - (-72)
-9d = 192
d = -192/9
d = -21.33 (rounded to 2 decimal places)
Therefore, the common difference is approximately -21.33.
To find the common difference of an arithmetic progression (AP), we need to find the difference between any two consecutive terms.
Let's denote the first term of the AP as "a" and the common difference as "d."
Given that the 4th term is 60, we can represent it in terms of "a" and "d" as:
a + 3d = 60 ----(1)
Also, given that the sum of the 8th and 9th terms is -72, we can represent it in terms of "a" and "d" as:
(a + 7d) + (a + 8d) = -72
2a + 15d = -72 ----(2)
We now have a system of two equations with two variables. We can solve this system of equations to find the values of "a" and "d."
Step 1: Multiply equation (1) by 2
2(a + 3d) = 2(60)
2a + 6d = 120 ----(3)
Step 2: Subtract equation (2) from equation (3)
2a + 6d - 2a - 15d = 120 - (-72)
-9d = 192
d = -192/9
d = -64/3
Therefore, the common difference is -64/3.
To find the common difference of an arithmetic progression (A.P.), we can use the following formula:
nth term = a + (n - 1)d
Where:
- nth term is the term number we want to find in the A.P.
- a is the first term of the A.P.
- n is the term number we want to find.
- d is the common difference.
In this case, we are given that the 4th term of the A.P. is 60. Let's use this information to find the value of 'a' using the formula:
4th term = a + (4 - 1)d
60 = a + 3d ...(1)
We are also given that the sum of the 8th and 9th terms is -72. Using this information, we can find the value of a + 7d (the 8th term) and a + 8d (the 9th term). Then we'll use these values to form an equation:
8th term + 9th term = a + 7d + a + 8d
-72 = 2a + 15d ...(2)
Now, we have a system of two linear equations (equation 1 and equation 2) with two variables (a and d). We can solve this system of equations to find the common difference (d).
Let's solve the equations:
Equation 1: 60 = a + 3d ...(1)
Equation 2: -72 = 2a + 15d ...(2)
Multiply equation 1 by 2:
2 * (60 = a + 3d) -> 120 = 2a + 6d ...(3)
Now, subtract equation 3 from equation 2:
-72 - 120 = 2a + 15d - (2a + 6d)
-192 = 9d
d = -192/9
d = -21.333...
So, the common difference is approximately -21.333 (or -21.33 when rounded to two decimal places).