The 4th term of an arithmetic progression is 6.if the sum of the 8th and 9th terms is -72.what is the common difference
6 + 4d + 6 + 5d = -72
12 + 9d = -72 ... 9d = -84 ... d = -9 1/3
To find the common difference in an arithmetic progression (AP), we need at least two terms.
Given:
- The 4th term is 6.
- The sum of the 8th and 9th terms is -72.
Let's find the common difference step-by-step:
Step 1: Find the difference between the 4th term and the 1st term.
Since we don't know the 1st term, let's assume it as "a". The 4th term is given as 6. The common difference is denoted as "d". Therefore, we can express the 4th term as: a + 3d = 6.
Step 2: Find the difference between the 9th term and the 8th term.
The 8th term is represented as (a + 7d). The 9th term is represented as (a + 8d).
According to the given information, the sum of the 8th and 9th terms is -72, so we have the equation: (a + 7d) + (a + 8d) = -72. Simplify this equation: 2a + 15d = -72.
Step 3: Solve the two equations simultaneously.
To solve these two equations simultaneously, we can use substitution or elimination methods. Let's use the elimination method:
Multiply the first equation by 2 to get: 2(a + 3d) = 12, which simplifies to: 2a + 6d = 12.
Write down both the modified equations:
2a + 6d = 12 ...(Equation 1)
2a + 15d = -72 ...(Equation 2)
Subtract Equation 1 from Equation 2:
(2a + 15d) - (2a + 6d) = -72 - 12.
This simplifies to: 9d = -84.
Step 4: Solve for 'd'.
Divide both sides of the equation by 9: 9d/9 = -84/9.
This gives us: d = -84/9 = -28/3.
Therefore, the common difference (d) of the arithmetic progression is -28/3 or approximately -9.33 (rounded to two decimal places).
Well, the common difference is a sneaky little thing. It's like that friend who always hides your snacks and hopes you won't notice. But fear not, I have the answer for you!
Let's solve this step by step. If the 4th term of the arithmetic progression is 6, we can assume the general term of the progression is given by:
a + (n - 1)d = 6
Now let's tackle the second part. The sum of the 8th and 9th terms is -72. Using the same logic, we can express this as:
[2a + (8-1)d] + [2a + (9-1)d] = -72
Simplifying that, we get:
2a + 7d + 2a + 8d = -72
4a + 15d = -72
Now we have 2 equations:
a + 3d = 6 (Equation 1)
4a + 15d = -72 (Equation 2)
To eliminate a, we can multiply Equation 1 by 4, then subtract it from Equation 2:
4a + 12d = 24
4a + 15d = -72
Subtracting, we get:
3d = -96
So, the common difference (d) is -32. It seems like it went on a diet of negative numbers!
To find the common difference (d) in an arithmetic progression, we can use the formula:
an = a1 + (n - 1)d
where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
Given information:
a4 = 6
a8 + a9 = -72
Step 1: Find the common difference (d) using the fourth term (a4).
Substitute a4 = 6 and n = 4 into the formula:
a4 = a1 + (4 - 1)d
6 = a1 + 3d
Step 2: Find the sum of the eighth and ninth terms (a8 + a9).
Substitute a8 + a9 = -72 into the formula:
a8 + a9 = a1 + (8 - 1)d + a1 + (9 - 1)d
-72 = 2a1 + 7d
Step 3: Equate the expressions for a8 + a9 derived in step 2 and solve for a1.
Since both expressions are equal to each other:
-72 = 2a1 + 7d
Rearrange the equation:
2a1 = -72 - 7d
2a1 = -72 - 7d
a1 = (-72 - 7d)/2
Step 4: Substitute the value of a1 from step 3 into the equation from step 1 and solve for d.
6 = a1 + 3d
Substitute a1 with (-72 - 7d)/2:
6 = (-72 - 7d)/2 + 3d
Multiply both sides by 2 to eliminate the fraction:
12 = -72 - 7d + 6d
12 = -72 - d
d = -12 - 72
d = -84
Therefore, the common difference (d) in the arithmetic progression is -84.