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Chapter 7: Problem 580

th term is constant term in expansion of \(\left[\left(3 / x^{2}\right)+(\sqrt{x} / 3)\right]^{10}, x \neq 0\) (a) 4 (b) 7 (c) 8 (d) 9

Short Answer

Expert verified
Since there is no constant term in the expansion, there is no specific 'th' term for the constant term in the expansion of \(\left[\left(3 / x^{2}\right)+(\sqrt{x} / 3)\right]^{10}\), with \(x \neq 0\).

Step by step solution

01

Review binomial expansion theorem

Recall the binomial expansion theorem which states the expansion of \((a+b)^n\) where n is a non-negative integer, is given by: \[(a+b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n} a^0 b^n\]
02

Apply binomial expansion to given expression

Applying the binomial expansion theorem to our given expression, we have: \[\left[\left(3 / x^{2}\right)+(\sqrt{x} / 3)\right]^{10} = \sum_{k=0}^{10} \binom{10}{k} \left(\frac{3}{x^2}\right)^{10-k} \left(\frac{\sqrt{x}}{3}\right)^{k}\]
03

Simplify the expression

Now, we need to simplify the expression by combining the terms: \[\sum_{k=0}^{10} \binom{10}{k} \left(\frac{3^{10-k}}{x^{20-2k}}\right) \left(\frac{x^k}{3^k}\right) = \sum_{k=0}^{10} \binom{10}{k} \frac{3^{10-2k} x^k}{x^{20-2k}}\]
04

Find the term with no x-variable

We want to find the term with no variable x, i.e., the constant term. So, we have to find the value of k for which the power of x becomes zero: \[20-2k+k = 0\] Solving for k, we get: \[k=20-2k\] \[3k=20\] \[k = \frac{20}{3}\] Since k should be an integer, there is no term with no variable x, which means there is no constant term.
05

Determine the th term

Based on our findings, we can conclude that the constant term doesn't exist in the expansion. Therefore, the th term is None.

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Most popular questions from this chapter

Co-efficient of \(\mathrm{x}^{4}\) in expansion of \(\left(1+\mathrm{x}+\mathrm{x}^{2}+\mathrm{x}^{3}\right)^{11}\) is (a) 330 (b) 990 (c) 1040 (d) 900

The coefficient of \((1 / x)\) in expansion of \((1+x)^{n}[1+(1 / x)]^{n}\) is (a) \({ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}}\) (b) \({ }^{2 \mathrm{n}} \mathrm{C}_{(\mathrm{n}-1)}\) (c) \((1 / 2)\) (d) \({ }^{2 \mathrm{n}} \mathrm{C}_{0}\)

The constant term in expansion of \(\left[\left\\{\left(3 \mathrm{x}^{2}\right) / 2\right\\}-(1 / 3 \mathrm{x})\right]^{9}\), \(\mathrm{x} \neq 0\) is (a) \((5 / 18)\) (b) \((7 / 18)\) (c) \((5 / 17)\) (d) \((7 / 17)\)

\({ }^{\mathrm{n}} \mathrm{C}_{1}+2 \cdot{ }^{\mathrm{n}} \mathrm{C}_{2}+3 \cdot{ }^{\mathrm{n}} \mathrm{C}_{3}+\ldots+\mathrm{n} \cdot{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}=\) (a) n \(\cdot 2^{n-1}\) (b) \((\mathrm{n}-1) 2^{\mathrm{n}-1}\) (c) \((\mathrm{n}+1) 2^{\mathrm{n}-1}\) (d) \((\mathrm{n}-1) 2^{\mathrm{n}}\)

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) \(210(3 / 2)^{4}\)
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