Find the nth term and first three terms of the arithemetic sequence having u4=93 and u8=65.
Answers
from your given:
a + 3d = 93
a + 7d = 65
subtract them:
4d = -28
d = -7
in a+3d = 93
a - 21 = 93
a = 114
You do the rest
To find the nth term of an arithmetic sequence, we use the formula: Un = a + (n - 1) * d, where Un is the nth term, a is the first term, n is the position of the term, and d is the common difference.
We are given two terms: u4 = 93 and u8 = 65. Let's denote the first term as a and the common difference as d.
Using the given information:
u4 = a + (4 - 1) * d = a + 3d = 93
u8 = a + (8 - 1) * d = a + 7d = 65
Now we can solve these two equations simultaneously to find the values of a and d.
From equation 1: a + 3d = 93
From equation 2: a + 7d = 65
Subtract equation 1 from equation 2:
(a + 7d) - (a + 3d) = 65 - 93
4d = -28
d = -7
Substituting the value of d into equation 1:
a + 3(-7) = 93
a - 21 = 93
a = 93 + 21
a = 114
Therefore, the first term (a) is 114 and the common difference (d) is -7.
Now let's find the nth term and the first three terms.
The nth term (Un) formula is: Un = 114 + (n - 1) * (-7)
The first three terms (n = 1, 2, 3) are:
u1 = 114 + (1 - 1) * (-7) = 114
u2 = 114 + (2 - 1) * (-7) = 107
u3 = 114 + (3 - 1) * (-7) = 100
Therefore, the nth term of the arithmetic sequence is Un = 114 + (n - 1) * (-7), and the first three terms are 114, 107, and 100.
To find the nth term of an arithmetic sequence, we need to determine the common difference and an initial term.
Given that u4 = 93 and u8 = 65, we can use these terms to find the common difference:
The formula for the nth term of an arithmetic sequence is given by: u(n) = a + (n-1)d,
where u(n) is the nth term, a is the first term, n is the position of the term, and d is the common difference.
So, let's plug in the values we have:
u4 = 93
a + (4 - 1)d = 93
a + 3d = 93 ----(1)
u8 = 65
a + (8 - 1)d = 65
a + 7d = 65 ----(2)
Now, we have a system of two equations (equation 1 and equation 2). We can solve this system to find the values of a and d. The solution will provide us with the first term and the common difference.
Subtracting equation 2 from equation 1, we get:
(a + 3d) - (a + 7d) = 93 - 65
-4d = 28
d = -7
Now that we have the value of d, we can substitute it back into equation 1 to find a:
a + 3(-7) = 93
a - 21 = 93
a = 114
So, the common difference (d) is -7, and the first term (a) is 114.
Now, let's find the first three terms of the sequence:
The sequence is given by: a, a + d, a + 2d
Plugging in the values, we get:
First term (u1) = a = 114
Second term (u2) = a + d = 114 - 7 = 107
Third term (u3) = a + 2d = 114 - 14 = 100
Therefore, the nth term of the arithmetic sequence is u(n) = 114 - 7(n-1), and the first three terms are 114, 107, and 100.