The 9th term of an Arithmetic Progression is 8 and the 4th term is 20. Find the first term and the common difference.
There are 5 terms, with a total difference of -12, so d = -12/5
a + 3(-12/5) = 20
a - 36/5 = 20
a = 136/5
So, the sequence is
136/5, 124/5, 112/5, 100/5=20, 88/5, 76/5, 64/5, 52/5, 40/5=8, ...
Well, I guess this arithmetic progression is really confused because its terms are all over the place. It starts with a high number and then suddenly drops down. It's like the terms couldn't agree on a pattern and decided to have a laugh. But hey, I'm here to help, so let's sort it out!
Let's assume that the first term of this hilarious arithmetic progression is "a" and the common difference is "d."
Given that the 4th term is 20, we can use the formula for the nth term of an arithmetic progression:
a + (n-1)d = term
Substituting the values we know:
a + (4-1)d = 20
a + 3d = 20 ---(1)
Now, the 9th term is 8. Let's use the same formula again:
a + (9-1)d = 8
a + 8d = 8 ---(2)
We have a system of equations (1) and (2) to solve, and we'll find the values for "a" and "d."
But before we do that, let me take a moment to appreciate the comedy here. It's like term number 4 is all confident, strutting around with its value of 20. Then, term number 9 comes along and says, "Hold my clown nose, I'm dropping down to 8, watch me go!" Truly an unpredictable arithmetic progression.
Alright, now back to business. Solving the system of equations (1) and (2), we get:
a + 3d = 20 ---(1)
a + 8d = 8 ---(2)
Subtracting equation (1) from equation (2):
(a + 8d) - (a + 3d) = 8 - 20
8d - 3d = -12
5d = -12
Dividing both sides by 5:
d = -12/5
And now we can substitute the value of "d" in equation (1) to find "a":
a + 3(-12/5) = 20
a - 36/5 = 20
a = 20 + 36/5
a = 100/5 + 36/5
a = 136/5
So the first term, "a," is 136/5 and the common difference, "d," is -12/5.
I hope this clears up the confusion in this arithmetic progression. It's a classic case of a pattern gone clown-wild!
To find the first term and the common difference of an arithmetic progression, we can use the following formula:
a_n = a_1 + (n-1)d,
where a_n is the nth term, a_1 is the first term, n is the position of the term in the sequence, and d is the common difference.
Given that the 9th term (a_9) is 8 and the 4th term (a_4) is 20, we can use these values to set up two equations and solve for the first term (a_1) and the common difference (d).
Equation 1: a_9 = a_1 + (9-1)d -----> 8 = a_1 + 8d
Equation 2: a_4 = a_1 + (4-1)d -----> 20 = a_1 + 3d
To solve these two equations simultaneously, we can subtract Equation 2 from Equation 1:
8 - 20 = (a_1 + 8d) - (a_1 + 3d)
-12 = 5d
Dividing by 5 on both sides:
d = -12/5
Substituting the value of d into Equation 2, we can solve for a_1:
20 = a_1 + 3(-12/5)
20 = a_1 - 36/5
Multiplying both sides by 5 to eliminate the denominator:
100 = 5a_1 - 36
Adding 36 to both sides:
136 = 5a_1
Dividing by 5 on both sides:
a_1 = 136/5
Therefore, the first term (a_1) is 136/5 and the common difference (d) is -12/5.