4x+1,2x+4 and x+3 are first three terms of arithmetic sequence determine the value of x
a = (2x+4) - (4x+1) = -2x + 3
and
a = (x+3) - (2x+4) = -x - 1
so
2x-3 = x+1
x = 4
To determine the value of x, we can use the fact that the three terms form an arithmetic sequence.
The common difference between consecutive terms in an arithmetic sequence is constant.
So, we can set up the following equations:
2nd term - 1st term = 3rd term - 2nd term
(2x + 4) - (4x + 1) = (x + 3) - (2x + 4)
Simplifying the equation:
2x + 4 - 4x - 1 = x + 3 - 2x - 4
Combine like terms:
-2x + 3 = -x - 1
Subtracting -x from both sides:
-2x + x + 3 = -1
Simplifying:
-x + 3 = -1
Subtracting 3 from both sides:
-x = -4
Multiplying by -1 to solve for x:
x = 4
Therefore, the value of x is 4.
To find the value of x in an arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence. The formula is:
an = a1 + (n-1)d
Where:
an is the nth term,
a1 is the first term,
n is the position of the term,
and d is the common difference.
Given the first three terms: 4x+1, 2x+4, and x+3, we can set up the equation using the formula by substituting the given terms:
a1 = 4x+1
a2 = 2x+4
a3 = x+3
Now, we can find the common difference, d, by subtracting the second term from the first term:
d = a2 - a1 = (2x+4) - (4x+1)
= 2x - 4x + 4 - 1
= -2x + 3
Next, we compare the first and second terms in terms of the common difference to make sure they follow the arithmetic sequence pattern:
a2 - a1 = (2x+4) - (4x+1) = -2x + 3
Now, we compare the second and third terms:
a3 - a2 = (x+3) - (2x+4) = -x - 1
If the common difference, d, is the same for both comparisons, we can set the two expressions equal to each other:
-2x + 3 = -x -1
Now, we can solve this equation for x:
-2x + x = -1 - 3
-x = -4
x = 4
Therefore, the value of x in the arithmetic sequence is 4.