The next term of the G.P. `x ,x^2+2,a n dx^3+10` is `(729)/(16)` b. `6` c. `0` d. `54`

Options

A. `729/16`
B. 6
C. 0
D. 54

Explanation

To find the next term of the geometric progression (G.P.) given by the terms \( x, x^2 + 2, x^3 + 10 \), we can follow these steps:

### Step 1: Identify the terms of the G.P.
The first three terms of the G.P. are:
- First term \( a = x \)
- Second term \( b = x^2 + 2 \)
- Third term \( c = x^3 + 10 \)

### Step 2: Set up the equation for the common ratio
In a G.P., the ratio of consecutive terms is constant. Therefore, we can set up the following equation:
\[ \frac{b}{a} = \frac{c}{b} \]
This translates to:
\[ \frac{x^2 + 2}{x} = \frac{x^3 + 10}{x^2 + 2} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ (x^2 + 2)^2 = x(x^3 + 10) \]

### Step 4: Expand both sides
Expanding both sides:
\[ (x^2 + 2)(x^2 + 2) = x(x^3 + 10) \]
This simplifies to:
\[ x^4 + 4x^2 + 4 = x^4 + 10x \]

### Step 5: Rearrange the equation
Rearranging the equation leads to:
\[ x^4 + 4x^2 + 4 - x^4 - 10x = 0 \]
This simplifies to:
\[ 4x^2 - 10x + 4 = 0 \]

### Step 6: Factor the quadratic equation
We can factor out a 2:
\[ 2x^2 - 5x + 2 = 0 \]

### Step 7: Use the quadratic formula to find the roots
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 2, b = -5, c = 2 \):
\[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot 2}}{2 \cdot 2} \]
\[ x = \frac{5 \pm \sqrt{25 - 16}}{4} \]
\[ x = \frac{5 \pm 3}{4} \]
This gives us two possible values for \( x \):
1. \( x = \frac{8}{4} = 2 \)
2. \( x = \frac{2}{4} = \frac{1}{2} \)

### Step 8: Calculate the fourth term for each value of \( x \)
**Case 1: \( x = 2 \)**
- First term \( a = 2 \)
- Second term \( b = 2^2 + 2 = 6 \)
- Common ratio \( r = \frac{b}{a} = \frac{6}{2} = 3 \)
- Fourth term \( = a \cdot r^3 = 2 \cdot 3^3 = 2 \cdot 27 = 54 \)

**Case 2: \( x = \frac{1}{2} \)**
- First term \( a = \frac{1}{2} \)
- Second term \( b = \left(\frac{1}{2}\right)^2 + 2 = \frac{1}{4} + 2 = \frac{9}{4} \)
- Common ratio \( r = \frac{b}{a} = \frac{\frac{9}{4}}{\frac{1}{2}} = \frac{9}{2} \)
- Fourth term \( = a \cdot r^3 = \frac{1}{2} \cdot \left(\frac{9}{2}\right)^3 = \frac{1}{2} \cdot \frac{729}{8} = \frac{729}{16} \)

### Step 9: Conclusion
The possible fourth terms are \( 54 \) and \( \frac{729}{16} \).