If 1/2,1/x,1/3 are successive terms of an A.p,show that

Question

2-x/x_3=2/3

Answer 1

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.