Find the sum of G.P. :
`0*3+0*03+0*003+0*0003+ . . . . . . .` to 8 terms.

Answer

To find the sum of the geometric progression (G.P.) given by the series \(0.3 + 0.03 + 0.003 + 0.0003 + \ldots\) up to 8 terms, we can follow these steps:

### Step 1: Identify the first term and number of terms
The first term \(a\) of the G.P. is:
\[ a = 0.3 \]
The number of terms \(n\) is:
\[ n = 8 \]

### Step 2: Find the common ratio
To find the common ratio \(r\), we can use the formula:
\[ r = \frac{a_2}{a_1} \]
where \(a_1 = 0.3\) and \(a_2 = 0.03\):
\[ r = \frac{0.03}{0.3} = \frac{3}{30} = \frac{1}{10} = 0.1 \]

### Step 3: Use the formula for the sum of the first \(n\) terms of a G.P.
The formula for the sum \(S_n\) of the first \(n\) terms of a G.P. is:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
Substituting the values of \(a\), \(r\), and \(n\):
\[ S_8 = \frac{0.3(1 - (0.1)^8)}{1 - 0.1} \]

### Step 4: Calculate \(r^n\)
Calculate \(0.1^8\):
\[ 0.1^8 = 0.00000001 \]
So,
\[ 1 - (0.1)^8 = 1 - 0.00000001 = 0.99999999 \]

### Step 5: Substitute into the sum formula
Now substitute back into the sum formula:
\[ S_8 = \frac{0.3(0.99999999)}{0.9} \]

### Step 6: Simplify the expression
Calculating the numerator:
\[ 0.3 \times 0.99999999 \approx 0.3 \]
Now divide by \(0.9\):
\[ S_8 \approx \frac{0.3}{0.9} = \frac{1}{3} \approx 0.3333 \]

### Final Answer
Thus, the sum of the G.P. up to 8 terms is approximately:
\[ S_8 \approx 0.3333 \]

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