If 1/2,1/x,1/3 are successive terms of an A.p,show that
2-x/x_3=2/3
To show that the terms 1/2, 1/x, and 1/3 form an arithmetic progression (AP), we need to determine the common difference between the terms.
Let's start by finding the difference between the second term and the first term.
The second term is 1/x, and the first term is 1/2. The formula to find the difference between two terms in an AP is:
Common difference (d) = Second term - First term
So, in this case, the difference would be:
d = (1/x) - (1/2)
We can simplify this by finding a common denominator:
d = (2 - x) / (2x)
Now, let's find the difference between the third term and the second term.
The third term is 1/3, and the second term is 1/x. Again, using the same formula as above, we have:
d = (1/3) - (1/x)
Simplifying this:
d = (x - 3) / (3x)
Since this is an arithmetic progression, the common difference (d) should be the same for the differences between any two terms. Therefore, we can equate the two expressions for d:
(2 - x) / (2x) = (x - 3) / (3x)
To solve this equation, we can cross-multiply:
3x(2 - x) = 2x(x - 3)
Expanding and simplifying:
6x - 3x^2 = 2x^2 - 6x
Bringing all terms to one side:
0 = 2x^2 - 6x + 6x - 3x^2
0 = -x^2
Since x^2 is a positive value, in order for the equation to hold true, x must be zero.
Therefore, substituting x = 0 into the given equation:
(2 - x) / (2 * 0_3) = 2/3
(2 - 0) / (0_3) = 2/3
2 / 0_3 = 2/3
2 = 2/3
Hence, we have successfully shown that if the terms 1/2, 1/x, and 1/3 are successive terms of an AP, then 2 - x / (0_3) is equal to 2/3.
To show that 2 - x/x_3 = 2/3, we need to find x_3 in terms of x and substitute it into the equation.
Given that 1/2, 1/x, and 1/3 are successive terms of an arithmetic progression, we can write:
1/2, 1/x, 1/3
To find the common difference (d) between these terms, we subtract each term from the following term:
1/x - 1/2 = (2 - x)/2x
1/3 - 1/x = (x - 3)/3x
Since these differences should be equal (as it is an arithmetic progression), we can set them equal to each other and solve for x:
(2 - x)/2x = (x - 3)/3x
Cross-multiplying gives:
3x(2 - x) = 2x(x - 3)
6x - 3x^2 = 2x^2 - 6x
5x^2 - 6x - 18 = 0
Factoring the quadratic expression, we have:
(x - 3)(5x + 6) = 0
Setting each factor equal to zero and solving for x:
x - 3 = 0 -> x = 3
5x + 6 = 0 -> x = -6/5
However, since x is a denominator in our given terms, it cannot be negative. We therefore take x = 3 as our solution.
Now, we need to find x_3. The third term in the given arithmetic progression is 1/3. We can write this term in terms of x:
1/3 = 1/x + (3 - 1)/(2x)
Simplifying this equation:
1/3 = 1/x + 2/2x
1/3 = 1/x + 1/x
1/3 = 2/x
x/3 = 2
x = 6
Substituting this value of x into the equation 2 - x/x_3 = 2/3:
2 - 6/x_3 = 2/3
Cross-multiplying:
3(2 - 6/x_3) = 2
Distributing:
6 - 18/x_3 = 2
Multiplying both sides by x_3:
6x_3 - 18 = 2x_3
Subtracting 2x_3 from both sides:
4x_3 - 18 = 0
Adding 18 to both sides:
4x_3 = 18
Dividing by 4:
x_3 = 18/4
Simplifying:
x_3 = 9/2
Therefore, we have shown that 2 - x/x_3 = 2/3.
To prove that 2 - x/x_3 = 2/3, we first need to identify the common difference (d) of the arithmetic progression.
We know that the terms 1/2, 1/x, 1/3 are in an arithmetic progression. Hence, the common difference between these terms will be the same.
To find the common difference, we subtract the second term from the first term and compare it to the difference between the third term and the second term.
The difference between the first and second terms is:
1/x - 1/2
The difference between the second and third terms is:
1/3 - 1/x
Since these two differences should be equal, we have:
1/x - 1/2 = 1/3 - 1/x
To simplify the equation, we need to get rid of the fractions. We can do this by multiplying both sides of the equation by 6x:
6x * (1/x - 1/2) = 6x * (1/3 - 1/x)
Simplifying further, we have:
6 - 3x = 2x - 2
Combine like terms:
6 + 2 = 3x + 2x
8 = 5x
Divide both sides of the equation by 5:
8/5 = x
Now that we have found the value of x, we can substitute it back into the equation and simplify:
2 - x/x_3 = 2/3
2 - (8/5)/(8/5)_3 = 2/3
2 - (8/5)/(24/5) = 2/3
2 - (8/5)*(5/24) = 2/3
2 - 2/3 = 2/3
(6/3) - (2/3) = 2/3
4/3 = 2/3
Thus, we have proven that 2 - x/x_3 = 2/3.