If the rth term in the expansion of \(\left( \dfrac{x}{3} - \dfrac{2}{x^2} \right)^{10}\)contains x4, then rth term is equal to
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Finding the Term with \(x^4\) in Binomial Expansion
We are asked to find the term that contains \(x^4\) in the expansion of \(\left( \dfrac{x}{3} - \dfrac{2}{x^2} \right)^{10}\). We need to determine which term in the expansion has \(x\) raised to the power of 4. The expansion is of the form \((a+b)^n\), where \(a = \dfrac{x}{3}\), \(b = -\dfrac{2}{x^2}\), and \(n = 10\).
General Term in Binomial Expansion
The general term, often denoted as the (r+1)th term, in the expansion of \((a+b)^n\) is given by the formula:
\(T_{r+1} = \binom{n}{r} a^{n-r} b^r\)
Here, \(r\) is an integer ranging from 0 to \(n\).
Applying the General Term Formula
Substitute the values from our expression into the general term formula:
\(a = \dfrac{x}{3}\)
\(b = -\dfrac{2}{x^2}\)
\(n = 10\)
So, the general term \(T_{r+1}\) is:
\(T_{r+1} = \binom{10}{r} \left(\dfrac{x}{3}\right)^{10-r} \left(-\dfrac{2}{x^2}\right)^r\)
Simplifying the Term to Find the Power of x
Let's simplify the expression to isolate the terms involving \(x\):
\(T_{r+1} = \binom{10}{r} \cdot \dfrac{x^{10-r}}{3^{10-r}} \cdot (-2)^r \cdot \dfrac{1}{(x^2)^r}\)
\(T_{r+1} = \binom{10}{r} \cdot \dfrac{x^{10-r}}{3^{10-r}} \cdot (-2)^r \cdot \dfrac{1}{x^{2r}}\)
Combine the powers of \(x\):
\(T_{r+1} = \binom{10}{r} (-2)^r \dfrac{1}{3^{10-r}} x^{10-r - 2r}\)
\(T_{r+1} = \binom{10}{r} (-2)^r 3^{r-10} x^{10-3r}\)
The power of \(x\) in the general term \(T_{r+1}\) is \(10-3r\).
Equating the Power of x to 4
We are looking for the term that contains \(x^4\). Therefore, we set the power of \(x\) equal to 4:
\(10 - 3r = 4\)
Solving for r
Now, we solve this linear equation for \(r\):
\(10 - 4 = 3r\)
\(6 = 3r\)
\(r = \dfrac{6}{3}\)
\(r = 2\)
The value \(r=2\) corresponds to the term \(T_{r+1}\) that contains \(x^4\).
Identifying the Term Number
The term containing \(x^4\) is the \(T_{r+1}\) term. Since we found \(r=2\), the term is \(T_{2+1} = T_3\).
The question asks for the "rth term" which contains \(x^4\), and the options suggest the term number (1st, 2nd, 3rd, etc.). Our calculation shows that the 3rd term contains \(x^4\).
Therefore, the rth term (where r represents the term number) is the 3rd term.
| r value | Term Number (Tr+1) | Power of \(x\) (\(10-3r\)) |
|---|---|---|
| 0 | T1 | \(10 - 3(0) = 10\) |
| 1 | T2 | \(10 - 3(1) = 7\) |
| 2 | T3 | \(10 - 3(2) = 4\) |
| 3 | T4 | \(10 - 3(3) = 1\) |
| ... | ... | ... |
From the table, we can see that when \(r=2\), the power of \(x\) is 4, and this corresponds to the T3 term, which is the 3rd term in the expansion.
Revision Table: Binomial Expansion Concepts
| Concept | Description | Formula Example (\((a+b)^n\)) |
|---|---|---|
| Binomial Theorem | Formula for expanding \((a+b)^n\). | \((a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r\) |
| General Term | The (r+1)th term in the expansion. | \(T_{r+1} = \binom{n}{r} a^{n-r} b^r\) |
| Binomial Coefficient | \(\binom{n}{r}\) represents the number of ways to choose r items from a set of n. | \(\binom{n}{r} = \dfrac{n!}{r!(n-r)!}\) |
| Power of a Power | When raising a power to another power, multiply the exponents. | \((x^m)^n = x^{m \cdot n}\) |
| Dividing Powers with Same Base | Subtract the exponents when dividing powers with the same base. | \(\dfrac{x^m}{x^n} = x^{m-n}\) |
Additional Information: Finding Specific Terms
Finding specific terms in a binomial expansion is a common application of the general term formula. Here are a few things to keep in mind:
- The index \(r\) in the formula \(T_{r+1}\) starts from 0. This means the first term is \(T_1\) (when \(r=0\)), the second term is \(T_2\) (when \(r=1\)), and so on. The (r+1)th term corresponds to the value of \(r\).
- When dealing with expressions involving fractions and negative exponents, carefully apply the rules of exponents, especially when combining terms with the same variable.
- The total number of terms in the expansion of \((a+b)^n\) is \(n+1\).
- To find a term independent of \(x\) (a constant term), you would set the power of \(x\) in the general term equal to 0 and solve for \(r\).
Understanding the general term formula is key to solving various problems related to binomial expansions, such as finding specific terms, coefficients, or the middle term(s).
Important Questions from Binomial Expansion
The value of \(^{47}C_4 + \displaystyle\sum_{r=1}^5 {^{52-r}C_3}\) is equal to:
For every integer n > 2 the sum of the expansions \(1 - {}^n{C_1} + {}^n{C_2} + - - - {( - 1)^n}{}.^n{C_n}\) is______
What is the coefficient of x101y99 in the expansion of (2x - 3y)200?
If $x = \frac{1}{4}$, then the greatest term in the expansion of $(2 + 3x)^{15}$ will be
The coefficient of x11 in the expansion of \({\left( {{x^2} -\frac{1}{x}} \right)^{10}}\) is:
