Boolean Algebra Quiz

Boolean Algebra Overview

• Boolean Algebra is a branch of mathematics that focuses on variables that have two distinct states: true or false.
• It uses numerical representation, where '1' signifies true and '0' signifies false.
• This form of algebra differs from elementary algebra, which can involve a wide range of numerical values.

Characteristics of Boolean Algebra

• It is a fundamental topic in logic, discrete mathematics, and computer science.
• Boolean Algebra operates under specific laws and operations, including AND, OR, and NOT.
• Essential for various applications, particularly in digital circuit design and computer programming.

Importance in Technology

• Forms the basis for binary systems, which are used in computer systems and digital electronics.
• Underpins logical reasoning and decision-making processes in algorithms and data structures.

Expansion and Binomial Theorem

• The expression being expanded is ((\frac{2a}{3} + \frac{b}{a^2})^{8n}).
• The expansion utilizes the binomial theorem, which gives the general term as: [ T_k = \binom{n}{k-1} (A)^{(n-k+1)} (B)^{(k-1)} ] where (n) is the exponent, (A) is the first term, and (B) is the second term.

Terms in the Expansion

• The middle term for an expansion of (T^{8n}) (where (n) is an integer) is located using the formula for the (k)-th term.
• The total number of terms in the expansion is (8n + 1). The middle term corresponds to the (\frac{8n + 2}{2}) term.

Finding the Value of n

• Setting up the equation based on the middle term being the ninth term: [ T_{4n + 1} = T_9 ]
• The equation simplifies to: [ 4n + 1 = 9 ]
• Solving for (n):
  • Rearranging gives (4n = 8).
  • Finally, dividing by 4 leads to (n = 2).

Conclusion

• The answer to the problem is (n = 2).