The fourth term of and A.P is 6.if the sum of the 8th and 9th term is -72, Find the common difference
Let the common difference be represented by 'd'.
The fourth term of an A.P is given as a + 3d = 6 ...(1)
Here, a represents the first term.
The sum of the 8th and 9th term can be given as:
[(a + 7d) + (a + 8d)] = -72
2a + 15d = -72 ...(2)
Using equations (1) and (2), we can solve for 'd'.
Subtracting equation (1) from equation (2), we get:
2a + 15d - a - 3d = -72 - 6
a + 12d = -78 ...(3)
We can solve equations (1) and (3) to find the value of 'd'.
From equation (1):
a + 3d = 6
a = 6 - 3d ...(4)
Substituting equation (4) into equation (3):
6 - 3d + 12d = -78
9d = -84
d = -84/9
d = -28/3
Therefore, the common difference is -28/3.
Let's assume that the first term of the arithmetic progression is "a" and the common difference is "d".
Given that the fourth term is 6, we can write the equation for the fourth term as:
a + 3d = 6 ...(1)
We are also given that the sum of the 8th and 9th term is -72. Using the formula for the nth term of an arithmetic progression, we can write the equation as:
(a + 7d) + (a + 8d) = -72 ...(2)
Now, we can solve these two equations simultaneously to find the values of "a" and "d".
From equation (1), we have:
a = 6 - 3d
Substitute the value of "a" in equation (2), we get:
(6 - 3d + 7d) + (6 - 3d + 8d) = -72
(6 + 4d) + (6 + 5d) = -72
12 + 9d = -72
Subtract 12 from both sides of the equation:
9d = -84
Divide both sides of the equation by 9:
d = -84/9
d = -9.33 (approximately)
Therefore, the common difference (d) is approximately -9.33.
To find the common difference of an arithmetic progression (AP), we need to use the formula for the nth term of an AP.
For any arithmetic progression, the nth term is given by the formula:
\[ a_n = a_1 + (n-1)d \]
where:
- a_n is the nth term of the AP
- a_1 is the first term of the AP
- d is the common difference of the AP
- n is the position of the term in the AP
In this case, we are given that the fourth term (a_4) is 6. We can substitute the values into the formula to get:
\[ 6 = a_1 + (4-1)d \]
Simplifying the equation:
\[ 6 = a_1 + 3d \]
Now, we need to use the second piece of information given. The sum of the 8th and 9th terms of the AP is -72.
The sum of two consecutive terms in an AP can be found using the formula:
\[ S = \frac{n}{2}(a + l) \]
where:
- S is the sum of the terms
- n is the number of terms (in this case, 2)
- a is the first term of the two consecutive terms
- l is the last term of the two consecutive terms
In this case, we have the sum of the 8th and 9th terms as -72.
\[ -72 = \frac{2}{2}(a_8 + a_9) \]
\[ -72 = a_8 + a_9 \]
Now, we need to express these terms using the formula for the nth term we derived earlier.
For the 8th term:
\[ a_8 = a_1 + (8-1)d \]
\[ a_8 = a_1 + 7d \]
For the 9th term:
\[ a_9 = a_1 + (9-1)d \]
\[ a_9 = a_1 + 8d \]
Substituting these expressions into the sum equation:
\[ -72 = (a_1 + 7d) + (a_1 + 8d) \]
\[ -72 = 2a_1 + 15d \]
Now, we have a system of two equations:
\[ \begin{cases} 6 = a_1 + 3d \\ -72 = 2a_1 + 15d \end{cases} \]
We can solve this system of equations to find the values of a_1 and d.