The 4th term of an AP is 6,if the sum of the 8th&9th term is -72, what is the common difference
Let's assume the first term of the arithmetic progression is a and the common difference is d.
The formula to find the nth term of an arithmetic progression is:
Tn = a + (n-1)d
The 4th term can be calculated using this formula:
T4 = a + (4-1)d = a + 3d
We're given that the 4th term is 6:
T4 = 6
Substituting the value of T4 in the equation, we get:
a + 3d = 6
Using the same formula, the 8th term is:
T8 = a + (8-1)d = a + 7d
And the 9th term is:
T9 = a + (9-1)d = a + 8d
We're given that the sum of the 8th and 9th term is -72:
T8 + T9 = -72
(a + 7d) + (a + 8d) = -72
2a + 15d = -72
We now have a system of two equations:
a + 3d = 6 (Equation 1)
2a + 15d = -72 (Equation 2)
We can solve this system using substitution or elimination method.
Let's use the substitution method:
From Equation 1, we can express a in terms of d:
a = 6 - 3d
Substituting the value of a in Equation 2, we get:
2(6 - 3d) + 15d = -72
12 - 6d + 15d = -72
9d = -84
d = -84/9
d = -8
The common difference is -8.
To find the common difference in an arithmetic progression (AP), you can use the formula for the nth term of an AP:
nth term = first term + (n - 1) * common difference
Given that the 4th term is 6, we can substitute the values into the formula:
6 = first term + (4 - 1) * common difference
6 = first term + 3 * common difference
Next, we are given that the sum of the 8th and 9th terms is -72. Using the same formula, we can write the equation for the 8th term and the equation for the 9th term:
8th term = first term + (8 - 1) * common difference
9th term = first term + (9 - 1) * common difference
The sum of these two terms is -72:
8th term + 9th term = -72
(first term + 7 * common difference) + (first term + 8 * common difference) = -72
Simplifying the equation:
2 * first term + (15 * common difference) = -72
Now, we have a system of two equations:
1) 6 = first term + 3 * common difference
2) 2 * first term + 15 * common difference = -72
Solving this system of equations will give us the values of the first term and the common difference.
To find the common difference of an arithmetic progression (AP), we need to use the given information.
Let's assume that the first term of the AP is "a" and the common difference is "d".
The general formula for the nth term of an AP is given by:
An = a + (n-1)d
Given that the 4th term (A4) is 6, we can substitute the values into the formula:
6 = a + (4-1)d
6 = a + 3d ---(Equation 1)
Similarly, the sum of the 8th and 9th term is -72. The sum of two terms of an AP can be calculated using the formula:
Sum = (n/2)(2a + (n-1)d)
So, for the 8th and 9th term, we have:
-72 = (8/2)(2a + (8-1)d) + (9/2)(2a + (9-1)d)
-72 = 4(2a + 7d) + 4(2a + 8d)
Simplifying the equation:
-72 = 8a + 28d + 8a + 32d
-72 = 16a + 60d --- (Equation 2)
Now, we can solve the system of equations (Equation 1 and Equation 2) to find the values of "a" and "d".
From Equation 1:
a = 6 - 3d
Substituting this value in Equation 2:
-72 = 16(6-3d) + 60d
-72 = 96 - 48d + 60d
-72 - 96 = 12d
-168 = 12d
Dividing both sides by 12:
d = -14
Therefore, the common difference of the arithmetic progression is -14.