Find the nth term and first three terms of the arithmetic squence having U4=93 and U8=65?
U8-U4 = 4d, so d = (65-93)/4 = -7
U1 = U4 - 3d = 93+21 = 114
Un = 114 + (n-1)(-7) = 121 - 7n
What about the substitution method
Sure, I'll give you the nth term and the first three terms. But first, let me address the names of these terms. Are they named after robots or aliens? "U4" and "U8" sound like they could be designations for extraterrestrial beings. Maybe "U4" is an enthusiastic umbrella-wielding robot, and "U8" is a friendly alien who loves to give hugs.
Now, let's get back to the arithmetic sequence. To find the nth term, we can use the formula:
Un = U1 + (n - 1) * d
Where Un represents the nth term, U1 is the first term, n is the position of the term, and d is the common difference between terms.
To find the common difference, we can use the given terms U4 and U8. So, U4 = U1 + (4 - 1) * d = U1 + 3d, and U8 = U1 + (8 - 1) * d = U1 + 7d.
We have two equations now:
U1 + 3d = 93 ---(1)
U1 + 7d = 65 ---(2)
By solving these two equations, we can find the values of U1 and d, which will allow us to calculate the nth term and the first three terms. But let me tell you a joke before we go any further:
Why don't scientists trust atoms?
Because they make up everything!
Alright, back to the math. Let's find the common difference first. Subtracting equation (1) from equation (2), we get:
( U1 + 7d ) - ( U1 + 3d ) = 65 - 93
4d = -28
d = -7
Now that we have the value of the common difference, we can substitute it back into one of the equations to find U1. Let's use equation (1):
U1 + 3d = 93
U1 + 3(-7) = 93
U1 - 21 = 93
U1 = 114
Now we can finally find the nth term using the formula I mentioned before:
Un = U1 + (n - 1) * d
Substituting the values we found, the nth term becomes:
Un = 114 + (n - 1) * (-7)
And for the first three terms:
U1: 114
U2: 114 + (2 - 1) * (-7) = 107
U3: 114 + (3 - 1) * (-7) = 100
I hope that was helpful and made you smile!
To find the nth term of an arithmetic sequence, we need the formula:
Un = U1 + (n-1)d
where Un represents the nth term, U1 represents the first term, n represents the position of the term, and d represents the common difference.
Given that U4 is 93 and U8 is 65, we can use these values to find the common difference:
U4 = U1 + (4-1)d
93 = U1 + 3d (equation 1)
U8 = U1 + (8-1)d
65 = U1 + 7d (equation 2)
Now, we can solve these two equations simultaneously to find U1 and d.
Subtract equation 1 from equation 2:
65 - 93 = (U1 + 7d) - (U1 + 3d)
-28 = 4d
Divide both sides by 4:
-28/4 = d
-7 = d
Now, substitute the value of d back into equation 1 to find U1:
93 = U1 + 3(-7)
93 = U1 - 21
U1 = 93 + 21
U1 = 114
Therefore, the first term (U1) is 114 and the common difference (d) is -7.
To find the nth term, we can substitute the values of U1 and d into the formula:
Un = 114 + (n-1)(-7)
Un = 114 - 7n + 7
Un = 121 - 7n
Now, let's find the first three terms by substituting n values of 1, 2, and 3 into the formula:
U1 = 121 - 7(1) = 114
U2 = 121 - 7(2) = 107
U3 = 121 - 7(3) = 100
Therefore, the first three terms of the arithmetic sequence are 114, 107, and 100, respectively.
To find the nth term and the first three terms of an arithmetic sequence, you need to determine the common difference (d). The common difference is the constant value that is added to each term to get the next term in the sequence.
In this case, we are given U4 = 93 and U8 = 65. These represent the values of the fourth term and eighth term of the arithmetic sequence, respectively.
Using the formula for the nth term of an arithmetic sequence:
Un = U1 + (n - 1)d
For the fourth term (U4 = 93), we use n = 4:
93 = U1 + (4 - 1)d
93 = U1 + 3d
For the eighth term (U8 = 65), we use n = 8:
65 = U1 + (8 - 1)d
65 = U1 + 7d
Now, we have a system of two equations with two variables (U1 and d):
93 = U1 + 3d
65 = U1 + 7d
To solve the system, subtract the second equation from the first equation:
93 - 65 = (U1 + 3d) - (U1 + 7d)
28 = -4d
Divide both sides by -4:
28 / -4 = d
-7 = d
Now that we have the common difference, we can substitute it back into one of the equations to find U1:
65 = U1 + 7(-7)
65 = U1 - 49
U1 = 65 + 49
U1 = 114
Therefore, the common difference (d) is -7 and the first term (U1) is 114.
To find the nth term, use the formula:
Un = U1 + (n - 1)d
Substituting the values we found:
Un = 114 + (n - 1)(-7)
Un = 114 - 7n + 7
Un = 121 - 7n
Now you can find the first three terms by substituting n = 1, 2, and 3 into the formula:
U1 = 121 - 7(1)
U1 = 121 - 7
U1 = 114
U2 = 121 - 7(2)
U2 = 121 - 14
U2 = 107
U3 = 121 - 7(3)
U3 = 121 - 21
U3 = 100
So, the nth term of the arithmetic sequence is Un = 121 - 7n, and the first three terms are 114, 107, and 100.