Question

If b is the first term of an infinite $G.P$. whose sum is five, then b lies in the interval :

• A. $( - \infty , - 10] $
• B. $( - 10 , 0 ) $
• C. $(0 , 10)$
• D. $[10 , \infty)$

Answer 1

The Correct Option is C

To determine the interval in which \( b \) lies, let's analyze the problem given that \( b \) is the first term of an infinite Geometric Progression (G.P.) whose sum is 5.

The sum of an infinite G.P. with the first term \( b \) and common ratio \( r \) (where \( |r| < 1 \)) is given by the formula: 

\(S = \frac{b}{1 - r}\)

According to the problem, the sum \(S = 5\). So, we have:

\(\frac{b}{1 - r} = 5\)

Simplifying for \( b \), we find:

\(b = 5(1 - r)\)

Since it's an infinite G.P., the condition for it to converge is \(|r| < 1\). Hence the term \( 1 - r \) should be positive, implying:

\(1 > r\) and \(r > -1\)

For the simplicity of understanding, if we consider non-negative terms (a usual assumption unless stated otherwise), \( r \) would typically be between 0 and 1:

\(0 \leq r < 1\)

Thus:

\(b = 5(1 - r) \to 5 > b > 0\)

This means that \( b \) lies in the interval \((0, 10)\).

Let's verify why the other choices are incorrect:

• \((-\infty, -10]\): \( b \) cannot be negative in standard positive series definitions.
• \((-10, 0)\): For the sum to remain positive and sensible in common conditions of geometric progression, \( b \) cannot be negative.
• \([10, \infty)\): With \( 5 > b > 0 \), \( b \) cannot be greater than 10.

Therefore, the correct answer is \((0 , 10)\).