If an infinite GP has the first term x and the sum 5, then which one of the following is correct ?

Options

A. `x lt -10`
B. `-10 lt x lt 10`
C. `0 lt x lt 10`
D. `x gt 10`

Answer

To solve the problem step by step, we need to analyze the given information about the infinite geometric progression (GP).

### Step 1: Understand the formula for the sum of an infinite GP
The sum \( S \) of an infinite geometric series with first term \( a \) and common ratio \( r \) (where \( |r| < 1 \)) is given by:
\[ S = \frac{a}{1 - r} \]

### Step 2: Substitute the known values
In this case, the first term \( a = x \) and the sum \( S = 5 \). Therefore, we can write:
\[ 5 = \frac{x}{1 - r} \]

### Step 3: Rearranging the equation
From the equation above, we can rearrange it to express \( x \) in terms of \( r \):
\[ x = 5(1 - r) \]

### Step 4: Express \( r \) in terms of \( x \)
Rearranging the equation gives us:
\[ x = 5 - 5r \implies 5r = 5 - x \implies r = 1 - \frac{x}{5} \]

### Step 5: Apply the condition for convergence
For the sum of the infinite GP to converge, the absolute value of the common ratio must be less than 1:
\[ |r| < 1 \]
Substituting the expression for \( r \):
\[ |1 - \frac{x}{5}| < 1 \]

### Step 6: Solve the inequality
We can break this absolute value inequality into two parts:
1. \( 1 - \frac{x}{5} < 1 \)
2. \( 1 - \frac{x}{5} > -1 \)

#### Part 1: Solve \( 1 - \frac{x}{5} < 1 \)
This simplifies to:
\[ 0 < \frac{x}{5} \implies x > 0 \]

#### Part 2: Solve \( 1 - \frac{x}{5} > -1 \)
This simplifies to:
\[ 2 > \frac{x}{5} \implies x < 10 \]

### Step 7: Combine the results
From both parts, we find that:
\[ 0 < x < 10 \]

### Conclusion
Thus, the range of \( x \) is \( (0, 10) \).

### Final Answer
The correct option is that \( x \) lies in the range \( (0, 10) \).

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