Let's analyze the binomial expansion of (p + qx)⁹ and find the ratio of the coefficients of its middle terms when expanded in ascending powers of x.

Understanding Binomial Expansion and Middle Terms

A binomial expansion of the form (a + b)ⁿ has (n+1) terms. The general term, often denoted as T_r+1, is given by the formula:

\( T_{r+1} = \binom{n}{r} a^{n-r} b^r \)

In the given expansion (p + qx)⁹:

• a = p
• b = qx
• n = 9

The total number of terms in the expansion is n + 1 = 9 + 1 = 10.

Since the number of terms (10) is an even number, there will be two middle terms. These middle terms are the \( (n/2 + 1) \)th term and the \( (n/2 + 1 + 1) \)th term when indexed from T₁. Alternatively, for an even number of terms (n+1), the middle terms are the \( \frac{n+1}{2} \)th and \( \left(\frac{n+1}{2} + 1\right) \)th terms.

For n=9, the middle terms are:

• The \( \frac{9+1}{2} = \frac{10}{2} = 5 \)th term.
• The \( \left(\frac{9+1}{2} + 1\right) = (5+1) = 6 \)th term.

Finding the General Term and Coefficients

The general term T_r+1 in the expansion of (p + qx)⁹ is:

\( T_{r+1} = \binom{9}{r} p^{9-r} (qx)^r \)

This can be written as:

\( T_{r+1} = \binom{9}{r} p^{9-r} q^r x^r \)

The coefficient of the term containing x^r is \( \binom{9}{r} p^{9-r} q^r \).

Calculating Coefficients of Middle Terms

The two middle terms are the 5th term (T₅) and the 6th term (T₆).

For the 5th term (T₅), we have r+1 = 5, which means r = 4.

The coefficient of the 5th term is \( \binom{9}{4} p^{9-4} q^4 = \binom{9}{4} p^5 q^4 \).

For the 6th term (T₆), we have r+1 = 6, which means r = 5.

The coefficient of the 6th term is \( \binom{9}{5} p^{9-5} q^5 = \binom{9}{5} p^4 q^5 \).

Ratio of Coefficients in Ascending Powers of x

When expanded in ascending powers of x, the terms appear in the order x⁰, x¹, ..., x⁹. The 5th term contains x⁴ (since for T_r+1, the power of x is r), and the 6th term contains x⁵.

The ratio of the coefficients of the middle terms in ascending powers of x is (Coefficient of 5th term) / (Coefficient of 6th term).

Ratio = \( \frac{\text{Coefficient of } T_5}{\text{Coefficient of } T_6} = \frac{\binom{9}{4} p^5 q^4}{\binom{9}{5} p^4 q^5} \)

We know that \( \binom{n}{k} = \binom{n}{n-k} \). Therefore, \( \binom{9}{4} = \binom{9}{9-4} = \binom{9}{5} \).

So, the combination terms cancel out in the ratio:

Ratio = \( \frac{p^5 q^4}{p^4 q^5} \)

Simplifying the terms with exponents:

\( \frac{p^5}{p^4} = p^{5-4} = p^1 = p \)

\( \frac{q^4}{q^5} = q^{4-5} = q^{-1} = \frac{1}{q} \)

Ratio = \( p \times \frac{1}{q} = \frac{p}{q} \)

Thus, the ratio of the coefficients of the middle terms in the expansion of (p + qx)⁹ is \( \frac{p}{q} \).