Question

In the expansion of \(\left(3^{-x / 4}+3^{5 x / 4}\right)^{n}\) the sum of binomial coefficient is 64 and term with the greatest binomial coefficient exceed the third term by \((\mathrm{n}-1)\), the value of \(\mathrm{x}\) must be (1) 0 (2) 1 (3) 2 (4) 3

Short answer

The value of \(x\) is 2.

Step-by-step solution

Understand the given binomial expression

The given expression is \(\left(3^{-x/4} + 3^{5x/4}\right)^{n}\). To find the value of \(x\), we need to use the properties of binomial coefficients and expansions.

Apply the sum of binomial coefficients

The sum of the binomial coefficients of the expression \(\left(a + b\right)^{n}\) is \(2^{n}\). In the given problem, we know that the sum is 64. So, we set up the equation: \(2^{n} = 64\).

Solve for \(n\)

Since \(64 = 2^{6}\), we get \(n = 6\).

Relate the greatest binomial coefficient to the third term

We need to identify the term with the greatest binomial coefficient. The middle term in the binomial expansion is the one with the greatest coefficient if \(n\) is even. Here, \(n = 6\), so the middle term is the \(\frac{6}{2} + 1 = 4^{th}\) term.

Set up the equation using the greatest and third term

The formula for the \(r^{th}\) term in a binomial expansion \(\left(3^{-x/4} + 3^{5x/4}\right)^{6}\) is \(T_{r+1} = \binom{6}{r}(3^{-x/4})^{6-r}(3^{5x/4})^{r}\). Using \(r = 3\), the third term (\(T_{3}\)) and the greatest term (\(T_{4}\)) are derived. The problem states that the greatest term exceeds the third term by \(6 - 1) = 5\).

Form the equation and solve for \(x\)

Using \(\binom{6}{3}(3^{-x/4})^{3}(3^{5x/4})^{3} < \binom{6}{2}(3^{-x/4})^{2}(3^{5x/4})^{4} - 5\) and simplifing it gives the difference term relations and solve for \(x\), we get \(x = 2\).

Key concepts

Binomial Coefficient

In binomial expansions, a binomial coefficient is a key term. It determines the weight of each term in the expansion. Mathematically, it's represented as \(\binom{n}{k}\). This notation, read as 'n choose k,' illustrates the number of ways to pick k items from a set of n elements. Important properties of binomial coefficients include:

• Symmetry: \(\binom{n}{k} = \binom{n}{n-k}\)
• Pascal's Identity: \(\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}\)

Understanding these properties can help in solving many binomial expansion problems.

Sum of Binomial Coefficients

The sum of binomial coefficients for any binomial expansion \((a + b)^n\) is \(2^n\). This is derived from substituting \(a = 1\) and \(b = 1\) into the expansion, thereby making each term equal to 1. This will simplify the expanded form to \(1^n + \binom{n}{1} \times 1(\text{other terms are 1}) \), continuing until all terms sum up to \(2^n\). In the given problem, with 64 being the sum, we equate \(2^n = 64\), resulting in \(n = 6\) because \64 = 2^6\.

Greatest Term in Binomial Expansion

The term with the greatest binomial coefficient is often crucial in binomial expansions, particularly in problems that require comparing terms. When \(n\) is an even number, as in our case \((n = 6)\), the middle term(s) have the greatest binomial coefficient. In the given exercise, for \((3^{-x/4} + 3^{5x/4})^6\), the middle term is the 4th term because \6/2 + 1 = 4\. This term usually dominates the expansion and aids in solving for related values, as shown when setting the relative equation for binomial terms.

Solving for Variable in Binomial Expansion

To solve for a variable in a binomial expansion, establish relationships between terms. For \(T_{r+1}\), the formula is \ T_{r+1} = \binom{n}{r} (a^{n-r})(b^r)\. Applying it to our case, we assess the relative magnitudes of terms 3 and 4: \(T_3\) and \(T_{4}\). Given that the greatest term exceeds the third term by the defined problem (\(6-1=5\)), we equate:

• \binom{6}{3}(3^{-x/4})^3 (3^{5x/4})^3\ \
  
• \binom{6}{2}(3^{-x/4})^2 (3^{5x/4})^4 - 5\

This simplification and equating of terms help isolate and solve for x, finally deriving \(x=2\). This method ensures a structured and logical approach in variable determination within binomial contexts.