Sequences & Series
Geometric Progression
Grade 11
Question:
<p>The first term of an infinite geometric progression is <i>x</i> and its sum is 5. Then</p>
<p>(A) \(0 \leq x \leq 10\)</p>
<p>(B) \(0 < x < 10\)</p>
<p>(C) \(-10 < x < 0\)</p>
<p>(D) \(x > 10\)</p>
Step-by-Step Solution
Key Concept: The sum of an infinite G.P. exists only when the common ratio has absolute value less than 1, which constrains the first term.
<p>For an infinite G.P. with first term <i>x</i> and common ratio <i>r</i>, the sum is \(S = \frac{x}{1-r} = 5\), which requires \(|r| < 1\) for convergence. This gives \(x = 5(1-r)\). Since \(-1 < r < 1\) and \(r \neq 1\), we have \(0 < x < 10\).</p>
Correct Answer: B