Question:

The ninth term of the expansion (3x−12x)8\left(3x-\frac {1}{2x}\right)^8 is

Updated On: Apr 25, 2024
  • −1512x9\frac {-1}{512x^9}
  • 1512x9\frac {1}{512x^9}
  • 1256.x8\frac {1}{256.x^8}
  • −1512x8\frac {-1}{512 x^8}
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The Correct Option is C

Solution and Explanation

The general term of the expansion (x+a)n(x + a)^n is
Tr+1=nCrxn−rarT_{r+1} =^nC_r x^{n-r} a^r .
We have (3x−12x)8\left(3x - \frac{1}{2x}\right)^{8}
Here, r=8,x=3x,a=(−12x),n=8 r=8, x =3x, a = \left(- \frac{1}{2x}\right), n=8
∴\therefore Nineth term T9=8C8(3x)8−8(−12x)8T_{9} =^{8}C_{8} \left(3x\right)^{8-8} \left(\frac{-1}{2x}\right)^{8}
=1256.x8= \frac{1}{256 .x^{8}}
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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.