Given the sequence: 4, x, 3x/2, ...
a. Find x if the sequence is arithmetic
b. Find x if the sequence is geometric
arithmetic
x - 4 = 3x/2 - x
2 x - 3x/2 = 4
4 x/2 - 3 x/2 = 4
x= 8
geometric
a = 4
a r = x
a r^2 = 3 x/2
so
4 r = x
4 r^2 = 3 x/2
4 r^2 = 3 (4 r) /2
4 r^2 = 6 r
2 r^2 = 3 r
2 r^2 - 3 r = 0
r (2 r - 3 ) = 0
r = 3/2
4, 6, 9
x = 6
or, for the geometric:
x/4 = (3x/2) / x = 3/2
2x = 12
x = 6
a. To determine the value of x if the sequence is arithmetic, we need to check if the differences between consecutive terms are constant.
The differences between the terms are as follows:
- The difference between the first and second term is x - 4.
- The difference between the second and third term is (3x/2) - x = (x/2).
For an arithmetic sequence, the differences between consecutive terms must be the same. Therefore, we can set up the following equation:
x - 4 = x/2
To solve for x, we can multiply both sides of the equation by 2 to eliminate the fraction:
2(x - 4) = x
2x - 8 = x
Rearranging the equation, we get:
x = 8
Therefore, if the sequence is arithmetic, the value of x is 8.
b. To determine the value of x if the sequence is geometric, we need to check if the ratios between consecutive terms are constant.
The ratios between the terms are as follows:
- The ratio between the first and second term is x/4.
- The ratio between the second and third term is (3x/2)/(x) = 3/2.
For a geometric sequence, the ratios between consecutive terms must be the same. Therefore, we can set up the following equation:
x/4 = 3/2
To solve for x, we can multiply both sides of the equation by 4 to eliminate the fraction:
4(x/4) = 4(3/2)
x = 12/2
Simplifying, we get:
x = 6
Therefore, if the sequence is geometric, the value of x is 6.
To find x for each case, we need to understand the properties of arithmetic and geometric sequences.
a. Arithmetic Sequence:
An arithmetic sequence is a sequence in which the difference between any two consecutive terms is constant. To find x in this case, we can compare the difference between consecutive terms.
Let's calculate the difference between each pair of consecutive terms in the sequence:
Second term - First term: x - 4
Third term - Second term: (3x/2) - x
In an arithmetic sequence, the difference between consecutive terms should be constant. So we have the equation:
x - 4 = (3x/2) - x
We can simplify this equation and solve for x:
x - (3x/2) = -4
(2x - 3x)/2 = -4
-x/2 = -4
-x = -4 * 2
-x = -8
Dividing both sides by -1, we get:
x = 8
Therefore, the value of x that makes the sequence arithmetic is x = 8.
b. Geometric Sequence:
A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant factor, known as the common ratio. To find x in this case, we can calculate the ratio between consecutive terms.
Let's calculate the ratios between each pair of consecutive terms in the sequence:
(x)/(4) = (3x/2)/(x)
(x)/(4) = 3/2
To solve for x, we can cross-multiply and solve the resulting equation:
2x = 4 * 3
2x = 12
x = 12/2
x = 6
Therefore, the value of x that makes the sequence geometric is x = 6.
To summarize:
a. If the sequence is arithmetic, x = 8.
b. If the sequence is geometric, x = 6.