Question

The greatest term in expansion of \((1+\mathrm{x})^{10}\) is \(\mathrm{x}=(2 / 3)\) (a) \(210(3 / 2)^{6}\) (b) \(210(2 / 3)^{6}\) (c) \(210(2 / 3)^{4}\) (d) \(210(3 / 2)^{4}\)

Short answer

(a) \(210(2/3)^{6}\)

Step-by-step solution

Find the general term of the binomial expansion

The general term of a binomial expansion (a+b)^{n} can be expressed as: \(T_{r+1} = {}^{n}C_{r}a^{n-r}b^{r}\) For this exercise, n=10, a=1, and b=x. So the general term of (1+x)^{10} is: \(T_{r+1} = {}^{10}C_{r}1^{10-r}x^{r}\) Simplifying, we have: \(T_{r+1} = {}^{10}C_{r}x^{r}\)

Substitute x = 2/3 in the general term and find the greatest term

Now that we have the general term, we will calculate the values of each term, \(T_{r+1}\), where r = 0, 1, 2, ..., 10, by substituting x = (2/3) in the general term: \(T_{r+1} = {}^{10}C_{r}(2/3)^{r}\) To find the greatest term, we can compare the ratio of consecutive terms in the sequence. The ratio of two consecutive terms, \(T_{r+1}\) and \(T_{r}\), can be expressed as: \(\dfrac{T_{r+1}}{T_{r}} = \dfrac{^{10}C_{r}(2/3)^{r}}{^{10}C_{r-1}(2/3)^{r-1}}\)

Find the largest term using the ratio of consecutive terms

Now, we will simplify the ratio of consecutive terms and find the value of r for which it is maximum: \(\dfrac{T_{r+1}}{T_{r}} = \dfrac{^{10}C_{r}(2/3)^{r}}{^{10}C_{r-1}(2/3)^{r-1}} = \dfrac{\frac{10!}{r!(10-r)!} (2/3)^{r}}{\frac{10!}{(r-1)!(11-r)!} (2/3)^{r-1}}\) Simplifying this expression, we get: \(\dfrac{T_{r+1}}{T_{r}} = \dfrac{(11-r)(2/3)}{r}\) For the term to be the greatest term in the sequence, the ratio \(\dfrac{T_{r+1}}{T_{r}} > 1\). We can solve the inequality to get: \(\dfrac{(11-r)(2/3)}{r} > 1\) Solving this inequality, we get r < 6.7 (which is approximately r = 6 since it has to be an integer).

Calculate the greatest term using the value of r

Now that we know that r = 6 is the value that results in the greatest term, we can calculate the value of the greatest term: \(T_{7} = {}^{10}C_{6}(2/3)^{6}\) \(T_{7} = 210 (2/3)^{6}\) Thus, the greatest term in expansion of \((1+\mathrm{x})^{10}\) when x = (2/3) is:

Answer

(a) \(210(3 / 2)^{6}\)

Key concepts

Binomial Theorem

The binomial theorem is a fundamental mathematical principle that allows us to expand expressions raised to a power, where the expression is a binomial, i.e., it consists of two terms. The classic example of a binomial is (a+b), and the binomial theorem provides a formula for expanding expressions such as (a+b)^{n}, where ^n is a non-negative integer.

General Term of Binomial Expansion

In any binomial expansion, the individual terms have a systematic pattern and can be calculated using a formula for the general term. The general term is often represented as T_{r+1}, which stands for the (r+1)^{th} term of the expansion of (a+b)^{n}. This term can be expressed as T_{r+1} = ^{n}C_{r}a^{n-r}b^{r}, where ^{n}C_{r} is a binomial coefficient and represents the number of ways to choose r elements from n elements, commonly known as combinations.

The general term is crucial when calculating a specific term in the expansion without fully expanding the entire expression. For example, to find the 7th term in the expansion of (1+x)^{10}, one would use the formula to find T_{7} directly, avoiding the necessity to expand all preceding six terms.

Greatest Term in Binomial Expansion

Identifying the greatest term in a binomial expansion involves analyzing the terms of the expansion to find the maximum value. As the terms of the expansion increase and then decrease, there comes a point where a term is larger than its neighbors on both sides. This is the greatest term.

To locate the greatest term mathematically, we compare the ratios of consecutive terms. If the ratio \(\frac{T_{r+1}}{T_r} > 1\), it indicates that the terms are still increasing. When this ratio becomes less than or equal to 1, we have found the term just after the greatest term, hence pinpointing the greatest term itself. This process usually involves calculating the term for which \(\frac{(n-r+1)}{r} > b/a\), with a and b being the binomial's components. Once found, we compute that term to obtain the greatest value in the expansion.