Find the sum to indicated number of terms in each of the geometric progressions in Questions 7 to 10 :
`x^(3),x^(5),x^(7),..." n terms "(if x nepm1)`

Explanation

To find the sum of the first \( n \) terms of the geometric progression given by \( x^3, x^5, x^7, \ldots \), we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \( a \) of the series is:
\[ a = x^3 \]

To find the common ratio \( r \), we can take the ratio of the second term to the first term:
\[ r = \frac{a_2}{a_1} = \frac{x^5}{x^3} = x^{5-3} = x^2 \]

### Step 2: Write the formula for the sum of the first \( n \) terms of a geometric series
The formula for the sum \( S_n \) of the first \( n \) terms of a geometric progression is given by:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

### Step 3: Substitute the values into the formula
Now we substitute \( a = x^3 \) and \( r = x^2 \) into the formula:
\[ S_n = x^3 \frac{(x^2)^n - 1}{x^2 - 1} \]

### Step 4: Simplify the expression
We can simplify the expression:
\[ S_n = x^3 \frac{x^{2n} - 1}{x^2 - 1} \]

### Final Result
Thus, the sum of the first \( n \) terms of the geometric progression is:
\[ S_n = \frac{x^3 (x^{2n} - 1)}{x^2 - 1} \]

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