Let the sum of the coefficients of the first three terms in the expansion of $\left(x-\frac{3}{x^2}\right)^n, x \neq 0 n \in N$, be $376$. Then the coefficient of $x^4$ is ______
Let the sum of the coefficients of the first three terms in the expansion of $\left(x-\frac{3}{x^2}\right)^n, x \neq 0 n \in N$, be $376$. Then the coefficient of $x^4$ is ______
Correct Answer: 405
Approach Solution - 1
Given Binomial ,
Sum of coefficients of first three terms
Now general term
Coefficient of
Approach Solution -2
Step 1: Expand the series
The general term in the expansion of the series is given by:
\[ nC_r \cdot x^{n-r} \cdot (-3)^{2r} = nC_r \cdot (-3)^r \cdot x^{n - 3r}. \]
Step 2: Sum of coefficients of the first three terms
The first three terms correspond to \(r = 0, 1, 2\). These terms are:
\[ T_0 = nC_0 \cdot x^n, \quad T_1 = nC_1 \cdot (-3) \cdot x^{n-3}, \quad T_2 = nC_2 \cdot 9 \cdot x^{n-6}. \]
The sum of the coefficients is:
\[ nC_0 - nC_1 \cdot 3 + nC_2 \cdot 9 = 376. \]
Step 3: Solve for \(n\)
To solve for \(n\), simplify the equation:
\[ 1 - 3n + \frac{n(n - 1)}{2} \cdot 9 = 376. \] Simplifying further: \[ 9n^2 - 27n - 752 = 0. \]
Now, solve the quadratic equation:
\[ n = 10. \]
Step 4: Coefficient of \(x^4\)
To find the coefficient of \(x^4\), we set \(n - 3r = 4\). This gives:
\[ r = \frac{n - 4}{3} = \frac{10 - 4}{3} = 2. \]
The coefficient is:
\[ nC_2 \cdot (-3)^2 = 10C_2 \cdot 9 = 45 \cdot 9 = 405. \]
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.