The 4th term of an arithmetic progression is 6. If the sum of the 8th and 9th term is -72, find the common difference
Let's denote the first term of the arithmetic progression as "a" and the common difference as "d".
We are given that the 4th term is 6, so we can write the equation:
a + 3d = 6 ...(1)
We are also given that the sum of the 8th and 9th term is -72, so we can write the equation:
(a + 7d) + (a + 8d) = -72
Simplifying the equation, we get:
2a + 15d = -72 ...(2)
We can solve this system of equations (equations 1 and 2) to find the values of "a" and "d".
Multiplying equation (1) by 2, we get:
2a + 6d = 12 ...(3)
Subtracting equation (3) from equation (2), we get:
(2a + 15d) - (2a + 6d) = -72 - 12
9d = -84
d = -84/9
d = -28/3
Therefore, the common difference is -28/3.
To find the common difference, we can use the formula for the nth term of an arithmetic progression:
\[a_n = a_1 + (n-1)d\]
where \(a_n\) is the nth term, \(a_1\) is the first term, \(n\) is the number of terms, and \(d\) is the common difference.
Given that the 4th term is 6, we can plug in the values into the formula:
\[6 = a_1 + (4-1)d\]
Simplifying, we get:
\[6 = a_1 + 3d\]
Now, let's find the sum of the 8th and 9th terms using the same formula:
\[a_8 + a_9 = a_1 + (8-1)d + a_1 + (9-1)d\]
Since the sum is -72, we can write the equation as:
\[-72 = 2a_1 + 16d\]
Now we have a system of two equations:
\[
\begin{align*}
6 &= a_1 + 3d \\
-72 &= 2a_1 + 16d \\
\end{align*}
\]
We can solve these equations simultaneously to find the values of \(a_1\) and \(d\).
Subtracting double of the first equation from the second equation, we get:
\[-72 - 2(6) = 2a_1 + 16d - 2(a_1 + 3d)\]
Simplifying, we have:
\[-84 = 10d\]
Dividing both sides by 10, we get:
\[-8.4 = d\]
Therefore, the common difference is -8.4.
To find the common difference of an arithmetic progression (AP), we need to use the formula:
nth term = a + (n - 1)d
where:
- nth term is the term we want to find,
- a is the first term of the AP, and
- d is the common difference.
Let's solve this step by step:
1. Given that the 4th term is 6, we can use the formula to solve for the equation:
6 = a + (4 - 1)d
6 = a + 3d -- (Equation 1)
2. Next, we are given that the sum of the 8th and 9th term is -72. We can use the formula for the sum of the terms of an AP:
Sum of n terms = (n/2)(2a + (n - 1)d)
Using this formula, we can write the equation for the sum of the 8th and 9th term:
-72 = (8/2)(2a + (8 - 1)d) + (9/2)(2a + (9 - 1)d)
-72 = 4(2a + 7d) + 4(2a + 8d)
-72 = 8a + 28d + 8a + 32d
-72 = 16a + 60d -- (Equation 2)
3. Now, we have two equations (Equations 1 and 2) with two unknowns (a and d). We can solve these simultaneous equations to find the values of a and d.
From Equation 1, we have:
a = 6 - 3d
Substitute this value of a in Equation 2:
-72 = 16(6 - 3d) + 60d
-72 = 96 - 48d + 60d
-72 - 96 = 12d
-168 = 12d
d = -168/12
d = -14
Therefore, the common difference of the arithmetic progression is -14.