Understanding the Expansion of (1/3) * (2/3)^6: Maximizing the Terms

In this article, we will explore the expansion of the expression (1/3) * (2/3)^6 using the binomial theorem, and specifically, we will determine the maximum term in this expansion. This process involves understanding the structure of the binomial expansion and identifying the term that contributes the most to the sum.

Introduction to the Binomial Theorem

The binomial theorem is a fundamental concept in algebra that provides a way to expand expressions of the form (a b)^n, where a and b are any numbers, and n is a positive integer. It states that:

(a b)^n Σ (from k0 to n) [C(n, k) * a^(n-k) * b^k]

where C(n, k) is the binomial coefficient, calculated as C(n, k) n! / [k! * (n-k)!].

Applying the Binomial Theorem to (1/3) * (2/3)^6

The given expression is (1/3) * (2/3)^6. To apply the binomial theorem, we first express it in the form (a b)^n. Here, it is:

(1/3) * (2/3)^6 (1/3) * (1/3 2/3)^6 (1/3)^1 * (1 2/3)^6

Step-by-Step Expansion Using the Binomial Theorem

Let's expand (1 2/3)^6 using the binomial theorem:

(1 2/3)^6 Σ (from k0 to 6) [C(6, k) * (1)^(6-k) * (2/3)^k]

Expanding this, we get:

C(6, 0) * (1)^(6-0) * (2/3)^0 1 C(6, 1) * (1)^(6-1) * (2/3)^1 6/3 2/1 C(6, 2) * (1)^(6-2) * (2/3)^2 15 * 4/9 60/9 20/3 C(6, 3) * (1)^(6-3) * (2/3)^3 20 * 8/27 160/27 C(6, 4) * (1)^(6-4) * (2/3)^4 15 * 16/81 240/81 80/27 C(6, 5) * (1)^(6-5) * (2/3)^5 6 * 32/243 192/243 64/81 C(6, 6) * (1)^(6-6) * (2/3)^6 1 * 64/729 64/729

Summing these terms, we get the expansion of (1 2/3)^6.

Now, we need to multiply each term by (1/3) to include the (1/3)^1 factor from the original expression:

(1/3) * 1 1/3 (1/3) * 2/1 2/3 (1/3) * 20/3 20/9 (1/3) * 160/27 160/81 (1/3) * 80/27 80/81 (1/3) * 64/81 64/243 (1/3) * 64/729 64/2187

Determining the Maximum Term in the Expansion

To find the maximum term in this expansion, we need to analyze the terms' values. The term with the highest coefficient will dominate the sum and be the maximum term.

The coefficients of the terms are:

1/3 2/3 20/9 160/81 80/81 64/243 64/2187

Comparing these, 80/81 is the largest coefficient.

Thus, the maximum term in the expansion is:

80/81 * (1/3)^1 80/243

Therefore, the maximum term in the expansion of (1/3) * (2/3)^6 is 80/243.

Conclusion

In summary, we successfully expanded the expression (1/3) * (2/3)^6 using the binomial theorem, and identified the term 80/243 as the maximum term in this expansion. This demonstration not only reinforces the application of the binomial theorem but also highlights the importance of analyzing coefficients for identifying the dominant term in an expansion.