Greatest term
Understanding the Greatest Term in Binomial Theorem
The Binomial Theorem is a fundamental result in algebra that provides a formula for expanding expressions of the form $(a + b)^n$, where $n$ is a non-negative integer. The theorem states that:
$$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$
Where $\binom{n}{k}$ is the binomial coefficient, which can be calculated as:
$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$
In this expansion, each term is of the form $\binom{n}{k} a^{n-k} b^k$ and is called a binomial term. The greatest term is the term with the largest value when the expansion is evaluated for specific values of $a$ and $b$.
Identifying the Greatest Term
To identify the greatest term in the expansion of $(a + b)^n$, we need to consider the values of $a$ and $b$ and their relative sizes. The greatest term is not necessarily the term with the largest exponent or coefficient; it depends on the numerical values of $a$ and $b$.
Factors Affecting the Greatest Term
- Value of $a$ and $b$: If $a > b$, the terms with higher powers of $a$ will generally be larger.
- Value of $n$: The index of the greatest term can change as $n$ increases.
- Ratio of $a$ to $b$: The ratio $\frac{a}{b}$ can help determine which term will be the greatest.
Formula for the Greatest Term
The greatest term can be found using the following approach:
- Calculate the ratio of successive terms in the binomial expansion.
- Find the value of $k$ for which the ratio changes from greater than 1 to less than 1.
The ratio of the $(k+1)$-th term to the $k$-th term is given by:
$$\frac{T_{k+1}}{T_k} = \frac{\binom{n}{k+1} a^{n-(k+1)} b^{k+1}}{\binom{n}{k} a^{n-k} b^k} = \frac{(n-k)(a)}{(k+1)(b)}$$
When this ratio is greater than 1, the terms are increasing. When it is less than 1, the terms are decreasing. The greatest term is the last term before the ratio drops below 1.
Examples
Let's consider the binomial expansion of $(2 + 3)^5$. To find the greatest term, we will compare the ratios of successive terms.
| Term ($T_k$) | Binomial Coefficient | $a^{n-k}$ | $b^k$ | Value of Term | Ratio $\frac{T_{k+1}}{T_k}$ |
|---|---|---|---|---|---|
| $T_0$ | $\binom{5}{0}$ | $2^5$ | $3^0$ | 32 | 2.5 |
| $T_1$ | $\binom{5}{1}$ | $2^4$ | $3^1$ | 160 | 1.6667 |
| $T_2$ | $\binom{5}{2}$ | $2^3$ | $3^2$ | 480 | 1.25 |
| $T_3$ | $\binom{5}{3}$ | $2^2$ | $3^3$ | 1080 | 0.8333 |
| $T_4$ | $\binom{5}{4}$ | $2^1$ | $3^4$ | 1620 | - |
| $T_5$ | $\binom{5}{5}$ | $2^0$ | $3^5$ | 243 | - |
From the table, we can see that the ratio drops below 1 when moving from $T_2$ to $T_3$. Therefore, the greatest term in the expansion of $(2 + 3)^5$ is $T_2$, which is 480.
Conclusion
The greatest term in a binomial expansion is not always obvious and requires careful analysis of the terms and their ratios. By understanding the relationship between the coefficients, powers of $a$ and $b$, and the binomial coefficients, one can determine the greatest term for any given binomial expression. This concept is particularly useful in probability, combinatorics, and other areas of mathematics where binomial expansions are applied.