Write down the first three terms of the binomial expansion \((1 + ax)^{n}\) in ascending powers of x. If the coefficients of x and x\(^{2}\) are 2 and \(\frac{3}{2}\) respectively, find the values of a and n.
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Correct Answer: Option n
Explanation:
\((1 + ax)^{n} = ^{n}C_{n} (1)^{n} + ^{n}C_{n - 1} (1)^{n - 1} (ax) + ^{n}C_{n - 2} (1)^{n - 2} (ax)^{2} + ...\)
= \(1 + nax + (\frac{n(n - 1)}{2})(ax)^{2} + ... \)
Given
\(an = 2 .... (1)\)
\(\frac{n^{2} - n}{2}) a^{2} = \frac{3}{2} ... (2)\)
\(\frac{a^{2}n^{2} - a^{2} n}{2} = \frac{3}{2}\)
\(\frac{(an)^{2} - a(an)}{2} = \frac{3}{2} ... (3)\)
From (1), \(an = 2\). (3) becomes
\(\frac{2^{2} - a(2)}{2} = \frac{3}{2}\)
\(\frac{4 - 2a}{2} = \frac{3}{2}\)
\(2 - 2a = \frac{3}{2} \implies 2a = \frac{1}{2}\)
\(\therefore a = \frac{1}{4}\)
Recall \(an = 2\)
\(\frac{n}{4} = 2 \implies n = 8\)
\((1 + ax)^{n} = ^{n}C_{n} (1)^{n} + ^{n}C_{n - 1} (1)^{n - 1} (ax) + ^{n}C_{n - 2} (1)^{n - 2} (ax)^{2} + ...\)
= \(1 + nax + (\frac{n(n - 1)}{2})(ax)^{2} + ... \)
Given
\(an = 2 .... (1)\)
\(\frac{n^{2} - n}{2}) a^{2} = \frac{3}{2} ... (2)\)
\(\frac{a^{2}n^{2} - a^{2} n}{2} = \frac{3}{2}\)
\(\frac{(an)^{2} - a(an)}{2} = \frac{3}{2} ... (3)\)
From (1), \(an = 2\). (3) becomes
\(\frac{2^{2} - a(2)}{2} = \frac{3}{2}\)
\(\frac{4 - 2a}{2} = \frac{3}{2}\)
\(2 - 2a = \frac{3}{2} \implies 2a = \frac{1}{2}\)
\(\therefore a = \frac{1}{4}\)
Recall \(an = 2\)
\(\frac{n}{4} = 2 \implies n = 8\)