If `a,a^(2)+2,a^(3)+10` be three consecutive terms of G.P., then the fourth term is

Options

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

Answer

To find the fourth term of the geometric progression (G.P.) given the first three terms \( a, a^2 + 2, a^3 + 10 \), we can follow these steps:

### Step 1: Understand the properties of a G.P.
In a G.P., the ratio of consecutive terms is constant. Therefore, we can express this relationship mathematically:
\[ \frac{a^2 + 2}{a} = \frac{a^3 + 10}{a^2 + 2} \]

### Step 2: Cross-multiply the terms
Cross-multiplying gives us:
\[ (a^2 + 2)^2 = a \cdot (a^3 + 10) \]

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

### Step 4: Rearrange the equation
Subtract \( a^4 \) from both sides:
\[ 4a^2 + 4 = 10a \]
Rearranging gives us:
\[ 4a^2 - 10a + 4 = 0 \]

### Step 5: Solve the quadratic equation
Using the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 4, b = -10, c = 4 \):
\[ a = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 4 \cdot 4}}{2 \cdot 4} \]
\[ = \frac{10 \pm \sqrt{100 - 64}}{8} \]
\[ = \frac{10 \pm \sqrt{36}}{8} \]
\[ = \frac{10 \pm 6}{8} \]
This gives us two possible values for \( a \):
\[ a = \frac{16}{8} = 2 \quad \text{and} \quad a = \frac{4}{8} = \frac{1}{2} \]

### Step 6: Find the common ratio \( r \)
For \( a = 2 \):
\[ r = \frac{a^2 + 2}{a} = \frac{2^2 + 2}{2} = \frac{4 + 2}{2} = 3 \]

For \( a = \frac{1}{2} \):
\[ r = \frac{a^2 + 2}{a} = \frac{(\frac{1}{2})^2 + 2}{\frac{1}{2}} = \frac{\frac{1}{4} + 2}{\frac{1}{2}} = \frac{\frac{1}{4} + \frac{8}{4}}{\frac{1}{2}} = \frac{\frac{9}{4}}{\frac{1}{2}} = \frac{9}{2} \]

### Step 7: Calculate the fourth term
The fourth term of a G.P. is given by:
\[ T_4 = a \cdot r^3 \]

For \( a = 2 \) and \( r = 3 \):
\[ T_4 = 2 \cdot 3^3 = 2 \cdot 27 = 54 \]

For \( a = \frac{1}{2} \) and \( r = \frac{9}{2} \):
\[ T_4 = \frac{1}{2} \cdot \left(\frac{9}{2}\right)^3 = \frac{1}{2} \cdot \frac{729}{8} = \frac{729}{16} \]

### Final Answer
Thus, the fourth term can be either \( 54 \) or \( \frac{729}{16} \).