Lesson Worksheet: General Term in the Binomial Theorem Mathematics

In this worksheet, we will practice finding a specific term and the coefficient of a specific term inside a binomial expansion without the need to fully expand the series.

Q1:

Find the third term in the expansion of 2𝑥+5𝑥.

  • A2,000𝑥
  • B200𝑥
  • C2,000𝑥
  • D200𝑥

Q2:

Find 𝑎 in the expansion of 5𝑥+𝑥5.

  • A10,500𝑥
  • B125𝑥
  • C84𝑥
  • D10,500𝑥
  • E125𝑥

Q3:

Determine the coefficient of 𝑥 in the expansion of 𝑥+1𝑥.

Q4:

The terms of the expansion of (2𝑥+𝑚𝑦) are arranged according to the descending powers of 𝑥. Given that 𝑎=2,560𝑥𝑦, find the value of 𝑚.

Q5:

If the coefficient of the third term in the expansion of 𝑥14 is 338, determine the middle term in the expansion.

  • A99512𝑥
  • B99512𝑥
  • C2311,024𝑥
  • D2311,024𝑥

Q6:

Consider the expansion of (8𝑥+2𝑦). Find the ratio between the eighth and the seventh terms, when written in descending powers of 𝑥.

  • A8𝑥𝑦
  • B28𝑥17𝑦
  • C28𝑦17𝑥
  • D17𝑦28𝑥
  • E17𝑥28𝑦

Q7:

Consider the expansion of (𝑎+𝑏), where 𝑎 is positive. Find the values of 𝑎, 𝑏, and 𝑛 given that 𝑇=215040, 𝑇=258048, and 𝑇=215040.

  • A𝑎=3, 𝑏=2, 𝑛=10
  • B𝑎=3, 𝑏=3, 𝑛=10
  • C𝑎=2, 𝑏=2, 𝑛=10
  • D𝑎=2, 𝑏=2, 𝑛=9

Q8:

Consider the binomial expansion of (3+7𝑥) in ascending powers of 𝑥. When 𝑥=6, one of the terms in the expansion is equal to twice its following term. Find the position of these two terms.

  • A𝑎, 𝑎
  • B𝑎, 𝑎
  • C𝑎, 𝑎
  • D𝑎, 𝑎

Q9:

Consider the expansion of (𝑚𝑥+8), where 𝑚 is a positive constant. Determine the values of 𝑚 and 𝑛, given that the ratio between the coefficients of 𝑎 and 𝑎 is equal to 6374,640 and that the ratio between the coefficients of 𝑎 and 𝑎 is equal to 491,360.

  • A𝑛=7, 𝑚=34
  • B𝑛=7, 𝑚=41
  • C𝑛=34, 𝑚=7
  • D𝑛=41, 𝑚=7

Q10:

Find the ratio between the fifteenth and seventeenth terms in the expansion of (𝑥12).

  • A𝑥12
  • B12𝑥
  • C12𝑥
  • D𝑥12

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