Binomial Theorem
Binomial Coefficients and AP
Grade 11
Question:
<p>If the coefficients of <i>p</i>th, (<i>p</i>+1)th and (<i>p</i>+2)th terms in the expansion of \((1 + x)^n\) are in AP, then</p>
<p>(a) \(n^2 - 2np + 4p = 0\)</p>
<p>(b) \(n^2 - n(4p + 1) + 4p^2 - 2 = 0\)</p>
<p>(c) \(n^2 - n(4p + 1) + 4p^2 = 0\)</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Use the condition that three terms are in AP: the middle term is the average of the first and third terms. Apply properties of binomial coefficients.
<p><strong>Solution:</strong> The coefficients of the <i>p</i>th, (<i>p</i>+1)th and (<i>p</i>+2)th terms are $\binom{n}{p-1}$, $\binom{n}{p}$, and $\binom{n}{p+1}$ respectively.</p><p>For AP: $2\binom{n}{p} = \binom{n}{p-1} + \binom{n}{p+1}$</p><p>Using the property of binomial coefficients and simplifying leads to $n^2 - n(4p + 1) + 4p^2 = 0$</p>
Correct Answer: C