Check whether the given sequences form a G.P. or not :
`(1)/(8),(1)/(24),(1)/(72),(1)/(216), . . . . . . .. . `

Answer

To determine whether the given sequence forms a Geometric Progression (G.P.), we need to check if the ratio of consecutive terms is constant. The given sequence is:

\[ \frac{1}{8}, \frac{1}{24}, \frac{1}{72}, \frac{1}{216}, \ldots \]

### Step 1: Identify the terms of the sequence
Let:
- \( a_1 = \frac{1}{8} \)
- \( a_2 = \frac{1}{24} \)
- \( a_3 = \frac{1}{72} \)
- \( a_4 = \frac{1}{216} \)

### Step 2: Calculate the common ratio \( r \)
The common ratio \( r \) of a G.P. is given by the formula:

\[ r = \frac{a_n}{a_{n-1}} \]

#### Calculate \( r \) for the first two terms:
\[ r_1 = \frac{a_2}{a_1} = \frac{\frac{1}{24}}{\frac{1}{8}} = \frac{1}{24} \times \frac{8}{1} = \frac{8}{24} = \frac{1}{3} \]

#### Calculate \( r \) for the second and third terms:
\[ r_2 = \frac{a_3}{a_2} = \frac{\frac{1}{72}}{\frac{1}{24}} = \frac{1}{72} \times \frac{24}{1} = \frac{24}{72} = \frac{1}{3} \]

#### Calculate \( r \) for the third and fourth terms:
\[ r_3 = \frac{a_4}{a_3} = \frac{\frac{1}{216}}{\frac{1}{72}} = \frac{1}{216} \times \frac{72}{1} = \frac{72}{216} = \frac{1}{3} \]

### Step 3: Compare the common ratios
From our calculations, we have:
- \( r_1 = \frac{1}{3} \)
- \( r_2 = \frac{1}{3} \)
- \( r_3 = \frac{1}{3} \)

Since all the common ratios are equal, we conclude that the sequence forms a G.P.

### Final Conclusion
Hence, the sequence \( \frac{1}{8}, \frac{1}{24}, \frac{1}{72}, \frac{1}{216}, \ldots \) forms a Geometric Progression.

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