Understanding the Binomial Expansion Problem

The problem asks us to find the coefficient \(a_r\) in the given series expansion:

\((1 + 2x + x^2)^n = \displaystyle\sum_{r = 0}^{2n} a_r x^r\)

This means that when the expression \((1 + 2x + x^2)^n\) is expanded in powers of \(x\), the coefficient of the term \(x^r\) is \(a_r\).

Simplifying the Expression

Let's look at the base of the expression, \((1 + 2x + x^2)\). This is a standard algebraic identity:

\(1 + 2x + x^2 = (1+x)^2\)

So, the original expression can be rewritten as:

\(((1 + x)^2)^n\)

Applying Laws of Exponents

Using the exponent rule \((a^m)^n = a^{m \times n}\), we can simplify the expression further:

\(((1 + x)^2)^n = (1+x)^{2n}\)

Now the problem is reduced to finding the coefficient of \(x^r\) in the expansion of \((1+x)^{2n}\).

Using the Binomial Theorem

The Binomial Theorem states that for any positive integer \(N\), the expansion of \((1+x)^N\) is given by:

\((1+x)^N = \binom{N}{0}x^0 + \binom{N}{1}x^1 + \binom{N}{2}x^2 + \dots + \binom{N}{N}x^N = \displaystyle\sum_{k=0}^{N} \binom{N}{k} x^k\)

Here, \(\binom{N}{k}\) represents the binomial coefficient, which is also denoted as \(^N C_k\).

Expanding \((1+x)^{2n}\)

In our case, \(N = 2n\). Applying the Binomial Theorem with \(N=2n\), we get the expansion of \((1+x)^{2n}\):

\((1+x)^{2n} = \binom{2n}{0}x^0 + \binom{2n}{1}x^1 + \binom{2n}{2}x^2 + \dots + \binom{2n}{2n}x^{2n}\)

This can be written in summation notation as:

\((1+x)^{2n} = \displaystyle\sum_{k=0}^{2n} \binom{2n}{k} x^k\)

Comparing Coefficients

We are given that the expansion is \(\displaystyle\sum_{r = 0}^{2n} a_r x^r\). We found that the expansion is also \(\displaystyle\sum_{k = 0}^{2n} \binom{2n}{k} x^k\).

Comparing the two summations:

• Given: \(\displaystyle\sum_{r = 0}^{2n} a_r x^r = a_0 x^0 + a_1 x^1 + a_2 x^2 + \dots + a_{2n} x^{2n}\)
• Calculated: \(\displaystyle\sum_{k = 0}^{2n} \binom{2n}{k} x^k = \binom{2n}{0} x^0 + \binom{2n}{1} x^1 + \binom{2n}{2} x^2 + \dots + \binom{2n}{2n} x^{2n}\)

For these two series to be equal for all values of \(x\), the coefficients of corresponding powers of \(x\) must be equal. Therefore, for the term with \(x^r\) (where \(r\) corresponds to \(k\)), the coefficient \(a_r\) must be equal to \(\binom{2n}{r}\).

So, \(a_r = \binom{2n}{r}\).

Final Result

Using the notation for combinations, \(\binom{N}{k} = ^N C_k\), we have:

\(a_r = ^{2n} C_r\)

Revision Table: Key Binomial Concepts

Additional Information: Properties of Binomial Coefficients

Binomial coefficients \(^n C_k\) have several interesting properties. Some of these include:

• Symmetry: \(^n C_k = ^n C_{n-k}\). This means the coefficient of \(x^k\) is the same as the coefficient of \(x^{n-k}\) in the expansion of \((1+x)^n\).
• Sum of coefficients: The sum of all coefficients in the expansion of \((1+x)^n\) is \(2^n\). This is found by setting \(x=1\) in the expansion: \((1+1)^n = \sum_{k=0}^n \binom{n}{k} (1)^k = \sum_{k=0}^n \binom{n}{k}\).
• Pascal's Identity: \(^n C_k = ^{n-1} C_{k-1} + ^{n-1} C_k\). This identity is the basis for constructing Pascal's Triangle, where each number is the sum of the two numbers directly above it.

Understanding these properties can be helpful when working with binomial expansions and related problems.