The 4th term of an AP is 6 if d sum of d 8th and 9th term is -72 the common difference is
Please type your question in proper English.
"d" is the common variable for the difference between consecutive terms of an AP and not as a substitute for "the".
Oh, I see you're asking about an arithmetic progression (AP). Well, let me entertain you with a funny response.
The common difference between terms in an arithmetic progression is like the distance between you and the exit sign in a maze. You just can't escape this mathematical twist!
To find the common difference, let's use our mathematical humor skills. Since we know that the 4th term of the AP is 6, let's call it the "joker" term. Why? Because it's always up for a good laugh!
Now, if the sum of the 8th and 9th term is -72, we're dealing with some serious comedy. It's like the 8th term told a joke that was so bad, the 9th term got depressed and made the sum negative!
But fear not, my friend, I'm here to solve this mystery. Let's call the common difference "Mr. Chuckles." Since we're looking for Mr. Chuckles, let's assume he's the same throughout the progression.
Using our mathematical prowess, we can set up the problem like this:
4th term + 3 * Mr. Chuckles = 6, and
8th term + 9th term = -72
But here's the punchline: With just these two equations, we can't solve for Mr. Chuckles directly. We need one more piece of information to unleash the comedic power of this arithmetic progression.
So until then, I'll leave you hanging with this cliffhanger. Keep laughing, my friend, and maybe Mr. Chuckles will reveal himself in time!
To find the common difference (d) of an Arithmetic Progression (AP), we can use the given information about the 4th, 8th, and 9th terms.
Let's use the formula for the nth term of an AP:
aₙ = a₁ + (n - 1)d
Given that the 4th term (a₄) is 6, we can substitute these values into the formula:
a₄ = a₁ + (4 - 1)d
6 = a₁ + 3d ----(1)
Now, we are given the sum of the 8th and 9th terms:
a₈ + a₉ = -72
Using the formula, we can substitute these values:
a₈ = a₁ + (8 - 1)d
a₉ = a₁ + (9 - 1)d
Substituting these into the sum equation:
(a₁ + 7d) + (a₁ + 8d) = -72
2a₁ + 15d = -72
Now we have two equations:
6 = a₁ + 3d ----(1)
2a₁ + 15d = -72 ----(2)
We can use these equations to solve for the common difference (d).
Multiplying equation (1) by 2, we get:
12 = 2a₁ + 6d ----(3)
Now, subtracting equation (3) from equation (2), we can eliminate a₁:
(2a₁ + 15d) - (2a₁ + 6d) = -72 - 12
9d = -84
Finally, solving for d:
d = -84 / 9
d = -9.3333 (rounded to 4 decimal places)
Therefore, the common difference (d) of the given arithmetic progression is approximately -9.3333.
To find the common difference of an arithmetic progression (AP), we can use the formula for the nth term of an AP, which is given by:
tn = a + (n-1)d
where:
tn is the nth term,
a is the first term, and
d is the common difference.
In this case, we have the 4th term, tn = 6. Let's substitute the values into the formula and solve for d:
6 = a + (4-1)d
6 = a + 3d
Now, we're also given that the sum of the 8th and 9th terms is -72. The sum of two terms of an AP can be calculated using the formula:
Sn = (n/2)(2a + (n-1)d)
where:
Sn is the sum of the first n terms.
Let's calculate the sum of the 8th and 9th terms, and substitute the value into the formula:
-72 = (8/2)(2a + (8-1)d) + (9/2)(2a + (9-1)d)
-72 = 4(2a + 7d) + 4(2a + 8d)
Now, we have two equations with two variables (a and d). We can solve these equations simultaneously to find the common difference.
Simplifying the equation:
-72 = 8a + 28d + 8a + 32d
-72 = 16a + 60d
Now, we have a system of linear equations:
6 = a + 3d
-72 = 16a + 60d
We can solve this system of equations using different methods such as substitution or elimination. Let's use substitution:
From the first equation, we have:
a = 6 - 3d
Substituting this value into the second equation:
-72 = 16(6-3d) + 60d
-72 = 96 - 48d + 60d
-72 = 96 + 12d
-12d = -72 - 96
-12d = -168
d = -168 / -12
d = 14
Therefore, the common difference of the arithmetic progression is 14.