Worksheet: General Term in the Binomial Theorem

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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 𝑎 in the expansion of 5√𝑥+√𝑥5.

  • A 1 2 5 𝑥
  • B 1 2 5 𝑥
  • C 8 4 𝑥
  • D 1 0 , 5 0 0 𝑥
  • E 1 0 , 5 0 0 𝑥

Q2:

Find the coefficient of 𝑎 in the expansion of (9𝑥+2).

Q3:

Find 𝑎 in the expansion of 𝑥+1𝑥.

  • A 𝑥
  • B 𝑥
  • C 3 6 4 𝑥
  • D 7 2 8 𝑥

Q4:

Find 𝑎 in the expansion of 24𝑥+𝑦4.

  • A 6 4 𝑥 𝑦
  • B 1 8 9 1 6 𝑥 𝑦
  • C 6 4 𝑥 𝑦
  • D 1 8 9 1 6 𝑥 𝑦

Q5:

Find the third term in the expansion of 𝑎+𝑏𝑎.

  • A 𝑎 𝑏
  • B 3 7 8 𝑎 𝑏
  • C 𝑎 𝑏
  • D 3 7 8 𝑎 𝑏

Q6:

Consider the binomial expansion of (2𝑥−𝑦) in ascending powers of 𝑥. What is the seventh term?

  • A 6 7 2 𝑥 𝑦
  • B − 6 7 2 𝑥 𝑦
  • C − 5 , 3 7 6 𝑥 𝑦
  • D 5 , 3 7 6 𝑥 𝑦

Q7:

Find the general term in 6𝑥−16𝑥.

  • A 𝐶 × 6 × 𝑥
  • B ( − 1 ) × 𝐶 × 6 × 𝑥
  • C ( − 1 ) × 𝐶 × 6 × 𝑥
  • D ( − 1 ) × 𝐶 × 6 × 𝑥
  • E ( − 1 ) × 𝐶 × 6 × 𝑥

Q8:

Find 𝑎 in the expansion of 9𝑥−1𝑥.

  • A ( − 1 ) × 𝐶 × 9 × 𝑥
  • B 𝐶 × 9 × 𝑥
  • C ( − 1 ) × 𝐶 × 9 × 𝑥
  • D ( − 1 ) × 𝐶 × 9 × 𝑥
  • E ( − 1 ) × 𝐶 × 9 × 𝑥

Q9:

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

  • A 2 , 0 0 0 𝑥
  • B 2 0 0 𝑥
  • C 2 , 0 0 0 𝑥
  • D 2 0 0 𝑥

Q10:

In a binomial expansion, where the general term is 𝐶𝑥, determine the position of the term containing 𝑥.

  • A 𝑎
  • B 𝑎
  • C 𝑎
  • D 𝑎

Q11:

Let 𝑎 be the 𝑘th term in the expansion of (1+𝑥) in increasing powers of 𝑥. Find all nonzero values of 𝑥 for which 2𝑎=𝑎+𝑎.

  • A2, 14
  • B2, 1
  • C2, −1
  • D4, 12

Q12:

Find the second-to-last term in (2+𝑥).

  • A 3 4 𝑥
  • B 6 8 𝑥
  • C 6 8 𝑥
  • D 3 4 𝑥

Q13:

If the ratio between the fourth term in the expansion of 𝑥+1𝑥 and the third term in the expansion of 𝑥−1𝑥 equals 7∶12, find the value of 𝑥.

  • A 4 9 1 , 2 9 6
  • B 1 , 2 9 6 4 9
  • C 7 3 6
  • D 3 6 7

Q14:

Consider the binomial expansion of (1+𝑥) in ascending powers of 𝑥. Given that 𝑎=𝑎 when 𝑥=1√5, find the value of 𝑛.

Q15:

Given that the sum of the first, middle, and last terms in the expansion of (𝑥−1) is 42,337, find all possible real values of 𝑥.

  • A 𝑥 = 2 1 6 , 𝑥 = − 1 9 6
  • B 𝑥 = 6 , 𝑥 = √ 1 9 6
  • C 𝑥 = − 6 , 𝑥 = − √ 1 9 6
  • D 𝑥 = 6 , 𝑥 = − √ 1 9 6
  • E 𝑥 = − 6 , 𝑥 = √ 1 9 6

Q16:

Let 𝑎 be the 𝑘th term in the expansion of (𝑥−2) in descending powers of 𝑥. Find all the nonzero values of 𝑥 for which 6𝑎−5𝑎+𝑎=0.

  • A 𝑥 = − 4 3 , 𝑥 = − 1 1 3
  • B 𝑥 = − 4 3 , 𝑥 = 2 3
  • C 𝑥 = − 1 1 3 , 𝑥 = 2 3
  • D 𝑥 = − 2 3 , 𝑥 = − 1 1 6

Q17:

Consider the expansion of (1+𝑥) in ascending powers of 𝑥. Given that the coefficient of 𝑥 is equal to the coefficient of 𝑎, determine the value of 𝑛.

Q18:

If the coefficient of the third term in the expansion of 𝑥−14 is 338, determine the middle term in the expansion.

  • A 2 3 1 1 , 0 2 4 𝑥
  • B 9 9 5 1 2 𝑥
  • C 9 9 5 1 2 𝑥
  • D 2 3 1 1 , 0 2 4 𝑥

Q19:

In the binomial expansion of (1+𝑥), 𝑛 is a positive, whole number and 𝑎 is the 𝑟th term, or the term which contains 𝑥.

If 8(𝑎)=27𝑎×𝑎, what is the value of 𝑛?

Q20:

Consider the expansion of (1+𝑥). Find the values of 𝑛 given that the coefficient of 𝑎 is equal to the coefficient of 𝑎.

Q21:

If 𝑎 is the term free of 𝑞 in 6𝑞−1𝑞, find 𝑛.

Q22:

Find 𝑎 in the expansion of 2√𝑥+√𝑥2.

  • A 3 9 , 9 3 6 𝑥
  • B 5 1 2 𝑥
  • C 7 8 𝑥
  • D 3 9 , 9 3 6 𝑥
  • E 5 1 2 𝑥

Q23:

Find 𝑎 in the expansion of 3𝑥−1𝑥.

  • A ( − 1 ) × 𝐶 × 3 × 𝑥
  • B ( − 1 ) × 𝐶 × 3 × 𝑥
  • C ( − 1 ) × 𝐶 × 3 × 𝑥
  • D ( − 1 ) × 𝐶 × 3 × 𝑥
  • E 𝐶 × 3 × 𝑥

Q24:

Find the coefficient of 𝑥 in the expansion of 1+𝑥−𝑥.

Q25:

Answer the following questions for the expansion of (2+4𝑥).

Given that the coefficient of 𝑥 is 3,840, find 𝑛.

  • A 𝑛 = 8
  • B 𝑛 = 5
  • C 𝑛 = 9
  • D 𝑛 = 7
  • E 𝑛 = 6

Hence, work out the value of the coefficient of 𝑥.

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