If the coefficients of x² and x³ in the expansion of (3 + ax)⁹ are the same, then the value of a is:

Question

Question

Answer 1

Concept:

General term: General term in the expansion of (a + b)n is given by

\(\rm {T_{\left( {r\; + \;1} \right)}} = \;{\;^n}{C_r} × {a^{n - r}} × {b^r}\)

Calculation:

We know that Tr+1 = ⁿCr an-r br.

In the given binomial expression (3 + ax)9, n = 9, a = 3 and b = ax.

∴ Tr+1 = 9Cr 39-r (ax)r = 9Cr 3⁹ \(\rm\left(\frac{a}{3}\right) ^r\) xr.

For the coefficients of x2 and x³, we must have r = 2 and 3 respectively.

⇒ 9C2 39 \(\rm\left(\frac{a}{3}\right) ^2\) = 9C3 39 \(\rm\left(\frac{a}{3}\right) ^3\)

⇒ a = \(\frac97\).