Consider the arithmetic sequence x+2, 18-x, 14+x,...
a) Find the value of x.
b) Find the general term of the arithmetic sequence.
in general an = a1 + (n-1)d
d = (18-x)-(x+2) = (14+x)-(18-x)
d = 16 -2x = -4 +2x
so
4 x = 20 and x = 5
then d = -4 + 2x = 6
so
an = a1 + (n-1)6
if the first term is x+2 = 7
then the sequence is
an = 7+(n-1)6
2,10,18,26
a1= 2
d=8
n=4
a5=2+(5-1)8
a5=2+32
a5=34
To find the value of x and the general term of the arithmetic sequence, we need to analyze the given sequence.
Let's write down the given sequence:
x + 2, 18 - x, 14 + x, ...
a) To find the value of x:
We can observe that each term is obtained by adding or subtracting x from a constant number. Since it is an arithmetic sequence, the difference between consecutive terms must be constant.
The difference between the first and second terms is 18 - x - (x + 2) = 16 - 2x.
The difference between the second and third terms is 14 + x - (18 - x) = 2x - 4.
Since the differences are supposed to be equal, we can set them equal to each other:
16 - 2x = 2x - 4
Simplifying the equation, we get:
4x = 20
Dividing both sides by 4, we find:
x = 5
So, the value of x is 5.
b) To find the general term of the arithmetic sequence:
To find the general term, we need to derive a pattern from the given sequence. We observe that each term can be written as a combination of x and a constant number.
We can rewrite the given sequence as:
x + 2, -x + 18, x + 14, ...
Looking closely, we can see that the constant number alternates between 2 and 18.
So, the general term can be written as:
a_n = (first term) + (n - 1) * (common difference)
The first term is x + 2, and the common difference is the difference between two consecutive terms, which is 18 - 2 = 16.
Therefore, the general term of the arithmetic sequence is:
a_n = (x + 2) + (n - 1) * 16
Now you can substitute the value of x with 5 to get the specific general term.