Given the sequence:4;x;32.Determine the value(s) of x if the sequence is:
1.arithmetic
2.geometric
AP:
x-4 = 32-x
2x=36
x=18
AP=4,18,32
GP:
x/4 = 32/x
x^2 = 128
x = ±√128
GP=4,√128,32
or
4,-√128,32
Arithmetic Progression - applying common difference,
$x-4=32-x$
$2x=32+4$
$2x=36$
$x= 18$
Geometric Progression - apply common ratio,
$\dfrac{x}{4} = \dfrac{32}{x}$
$x^2 =128$
$x = \pm \sqrt{128}$
1. If the sequence is arithmetic, we can find the common difference between the terms by subtracting consecutive terms.
Difference between 4 and x: x - 4
Difference between x and 32: 32 - x
Since it is an arithmetic sequence, the common difference should be the same. Therefore, we have the equation:
x - 4 = 32 - x
Simplifying this equation, we get:
2x = 36
Dividing by 2, we find:
x = 18
So, the value of x in the arithmetic sequence is 18.
2. If the sequence is geometric, we can find the common ratio by dividing consecutive terms.
Ratio between 4 and x: x/4
Ratio between x and 32: 32/x
Since it is a geometric sequence, the common ratio should be the same. Therefore, we have the equation:
x/4 = 32/x
Cross-multiplying, we get:
x^2 = 128
Taking the square root of both sides, we find:
x = √128
Now, the square root of 128 is approximately 11.314. However, it is important to note that the square root of negative numbers is not a real number, so we cannot have a negative value for x.
Therefore, in this case, there is no real value for x in the geometric sequence.
Hope that made you chuckle!
1. Arithmetic sequence:
In an arithmetic sequence, the difference between any two consecutive terms is constant. Let's find the common difference for the given sequence.
We have: 4, x, 32
The difference between the first and second terms is x - 4.
The difference between the second and third terms is 32 - x.
Since it is an arithmetic sequence, these differences should be equal. Therefore, we can equate them:
x - 4 = 32 - x
Simplifying the equation, we get:
2x = 36
Dividing both sides by 2:
x = 18
So, if the sequence is arithmetic, x = 18.
2. Geometric sequence:
In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio. To find the ratio, we can divide any term by the previous term.
We have: 4, x, 32
The ratio between the second and first terms is x/4.
The ratio between the third and second terms is 32/x.
Since it is a geometric sequence, these ratios should be equal. Therefore, we can equate them:
x/4 = 32/x
Cross-multiplying and simplifying the equation, we get:
x^2 = 128
Taking the square root of both sides:
x = ±√128
x = ±√(64 * 2)
x = ±(8 * √2)
So, if the sequence is geometric, x = 8√2 or x = -8√2.
To determine the value of x in the given sequence for both arithmetic and geometric sequences, we need to understand the properties of these two types of sequences.
1. Arithmetic Sequence:
In an arithmetic sequence, the difference between any two consecutive terms is always constant. To find the difference, we can subtract the first term from the second term or the second term from the third term since the difference remains the same for all terms.
Let's calculate the common difference:
Common difference = Second term - First term = 32 - 4 = 28
Now, we can find the value of x by adding the common difference to the first term or subtracting the common difference from the last term.
Value of x = First term + Common difference
Value of x = 4 + 28
Value of x = 32
Therefore, in an arithmetic sequence, the value of x is 32.
2. Geometric Sequence:
In a geometric sequence, each term is found by multiplying the previous term by a constant number called the common ratio. To find the common ratio, we can divide the second term by the first term or divide the third term by the second term, as the ratio remains the same for all terms.
Let's calculate the common ratio:
Common ratio = Second term / First term = 32 / 4 = 8
Now, we can find the value of x by multiplying the first term by the common ratio or dividing the last term by the common ratio.
Value of x = First term * Common ratio
Value of x = 4 * 8
Value of x = 32
Therefore, in a geometric sequence, the value of x is also 32.
In conclusion, for both an arithmetic and a geometric sequence, the value of x is 32 in the given sequence.