Find the term independent of x in the expansion of - x - 1 2- x 18 - x- 0

General term of the expansion is nC_ra^n rb^r=18C_rx^18 r3x^ r32^r given this term is independent of x so powers of x is zeroso 18 r3 r3=0= gt;r=9 t ...

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Standard XII

Mathematics

Greatest Binomial Coefficients

Find the term...

Question

Find the term independent of $x$ in the expansion of ${(\sqrt[3]{x} + \frac{1}{2 \sqrt[3]{x}})}^{18}, x > 0$

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Solution

general term of the expansion is $n C_{r} a^{n - r} b^{r} = 18 C_{r} x^{\frac{18 - r}{3}} \frac{x^{- r} 3}{2^{r}}$
given this term is independent of x so powers of x is zero
so $\frac{18 - r}{3} - \frac{r}{3} = 0 => r = 9$
therefore the term independent of x is $18 C_{9} (\frac{1}{2})^{9}$

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Find the term independent of x in the expansion of (3√x+123√x)18,x>0

general term of the expansion is nCran−rbr=18Crx18−r3x−r32rgiven this term is independent of x so powers of x is zeroso 18−r3−r3=0=>r=9therefore the term independent of x is 18C9(12)9

Find the term independent of $x$ in the expansion of ${(\sqrt[3]{x} + \frac{1}{2 \sqrt[3]{x}})}^{18}, x > 0$

general term of the expansion is $n C_{r} a^{n - r} b^{r} = 18 C_{r} x^{\frac{18 - r}{3}} \frac{x^{- r} 3}{2^{r}}$
given this term is independent of x so powers of x is zero
so $\frac{18 - r}{3} - \frac{r}{3} = 0 => r = 9$
therefore the term independent of x is $18 C_{9} (\frac{1}{2})^{9}$