If 5+x, 8, and 1+2x are consecutive terms in an arithmetic sequence, find x.
if arithmetic, then
8 - (5+x) = (1+2x) - 8
8 - 5 - x = 1 + 2x - 8
-3x = -10
x = 10/3
check:
terms are 25/3 , 8 , 23/2
or 25/3 , 24/3 , 23/3 which is an AS
Well, if 5+x, 8, and 1+2x are consecutive terms in an arithmetic sequence, then we can say that they are equally spaced.
But let's not jump to conclusions too quickly, because there's a reason why they call it an arithmetic sequence - it involves some arithmetic!
If we assume that 5 + x is the first term in the sequence, then the common difference would be (8 - (5 + x)) = (8 - 5 - x) = (3 - x).
Now, let's look at the second term, which is 8. The common difference is still (3 - x), so we can say that the third term (1 + 2x) is equal to the second term (8) plus the common difference.
So, 1 + 2x = 8 + (3 - x).
Simplifying this equation, we get:
1 + 2x = 8 + 3 - x.
Combine like terms:
2x + x = 11 - 1.
3x = 10.
Divide both sides by 3:
x = 10/3.
So, in this arithmetic sequence, x is equal to 10/3.
Now, that wasn't too bad, was it? Arithmetic can be quite fun when you add a little humor to it!
To find the value of x, we can use the formula for an arithmetic sequence which states that the common difference (d) between consecutive terms in an arithmetic sequence is the same.
The given terms are 5+x, 8, and 1+2x.
To find the common difference, we can subtract the second term from the first term and the third term from the second term:
(5+x) - 8 = 1+2x - 8
5 + x - 8 = 1 + 2x - 8
x - 3 = 2x - 7
Next, we can isolate the x-term by subtracting x from both sides:
x - x - 3 = 2x - x - 7
-3 = x - 7
Then, we can isolate the constant term on the right side by adding 7 to both sides:
-3 + 7 = x - 7 + 7
4 = x
Therefore, the value of x is 4.
To solve this problem, we need to define the arithmetic sequence and find the common difference.
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In this case, the given terms are 5+x, 8, and 1+2x. We need to determine the common difference, which is the same for all terms.
To find the common difference, we can compare any two consecutive terms. Let's compare the first two terms, 5+x and 8:
8 - (5+x) = 3 - x (Subtracting the first term from the second term)
Now, let's compare the second and third terms, 8 and 1+2x:
1+2x - 8 = -7 - 2x (Subtracting the second term from the third term)
Since we know that the common difference is the same for both comparisons, we can equate the two expressions:
3 - x = -7 - 2x
To solve for x, we need to simplify and isolate the x terms:
3 + 7 = -2x + x
10 = -x
Multiplying both sides of the equation by -1:
-10 = x
Hence, x is equal to -10.