Question

If the 4th, 10th, and 16th terms of a G.P. are \(x\), \(y\), and \(z\) respectively, then

• A. \(y = \sqrt{xz}\)
• B. \(x = \sqrt{yz}\)
• C. \(y = \frac{x + z}{2}\)
• D. \(z = \sqrt{xy}\)

Hint:

In a geometric progression, the nth term is given by \(ar^{n-1}\), where \(a\) is the first term and \(r\) is the common ratio.

Answer 1

The Correct Option is A

If the 4th, 10th, and 16th terms of a G.P. are \(x\), \(y\), and \(z\) respectively, then:

Step 1: General formula for the nth term of a G.P.
The general formula for the nth term of a geometric progression (G.P.) is given by: \[ T_n = ar^{n-1} \] where \( a \) is the first term and \( r \) is the common ratio of the G.P.

Step 2: Express the 4th, 10th, and 16th terms in terms of \( a \) and \( r \)
- The 4th term, \( T_4 = x \), is given by: \[ T_4 = ar^{4-1} = ar^3 = x \] - The 10th term, \( T_{10} = y \), is given by: \[ T_{10} = ar^{10-1} = ar^9 = y \] - The 16th term, \( T_{16} = z \), is given by: \[ T_{16} = ar^{16-1} = ar^{15} = z \]

Step 3: Set up a system of equations
From the above, we have the following equations: \[ ar^3 = x \quad \text{(Equation 1)} \] \[ ar^9 = y \quad \text{(Equation 2)} \] \[ ar^{15} = z \quad \text{(Equation 3)} \]

Step 4: Solve the system of equations
To find relationships between \( x \), \( y \), and \( z \), divide Equation 2 by Equation 1: \[ \frac{ar^9}{ar^3} = \frac{y}{x} \quad \Rightarrow \quad r^6 = \frac{y}{x} \quad \text{(Equation 4)} \] Now, divide Equation 3 by Equation 2: \[ \frac{ar^{15}}{ar^9} = \frac{z}{y} \quad \Rightarrow \quad r^6 = \frac{z}{y} \quad \text{(Equation 5)} \] From Equations 4 and 5, we get: \[ \frac{y}{x} = \frac{z}{y} \] Simplifying: \[ y^2 = xz \]

Step 5: Conclusion
Therefore, the relationship between \( x \), \( y \), and \( z \) is: \[ y^2 = xz \] This is the required result.