Sequences & Series
Arithmetic and Geometric Progressions
Grade 11
Question:
<p>Let <i>S</i> be the sum of the first 9 terms of the series <span style='display:inline-block;'>{<i>x</i> + <i>ka</i>} + {<i>x</i><sup>2</sup> + (<i>k</i> + 2)<i>a</i>} + {<i>x</i><sup>3</sup> + (<i>k</i> + 4)<i>a</i>} + {<i>x</i><sup>4</sup> + (<i>k</i> + 6)<i>a</i>} + ...</span> where <i>a</i> ≠ 0 and <i>x</i> ≠ 1.</p><p>If \(S = \frac{x^{10} - x + 45a(x-1)}{x-1}\), then <i>k</i> is equal to</p>
<p>(a) -5</p>
<p>(b) 1</p>
<p>(c) -3</p>
<p>(d) 3</p>
Step-by-Step Solution
Key Concept: Split the series into sum of powers of x and sum of arithmetic progression. Use formulas for geometric series sum and arithmetic series sum.
<p><strong>Solution:</strong></p><p>It is given that</p><p>$S = \{x + ka\} + \{x^2 + (k+2)a\} + \{x^3 + (k+4)a\} + \{x^4 + (k+6)a\} + \ldots\text{ up to 9 terms}$</p><p>$S = \{x + x^2 + x^3 + x^4 + \ldots + x^9\} + a\{k + (k+2) + (k+4) + \ldots + (k+16)\}$</p><p>$= \frac{x(x^9-1)}{x-1} + a\left(\frac{9}{2}\right)\{2k + (9-1) \cdot 2\}$</p><p>Comparing with the given expression and solving, we get <i>k</i> = -3.</p><p>∴ Answer is (c).</p>
Correct Answer: C