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 the third term in the expansion of ( 4 𝑥 + 3 ) 3 .

  • A 1 0 8 𝑥 2
  • B 2 7 𝑥
  • C 2 7 𝑥 2
  • D 1 0 8 𝑥

Q2:

Find 𝑇 4 in the expansion of 5 √ 𝑥 + √ 𝑥 5 9 .

  • A 1 2 5 𝑥 − 3 2
  • B 1 0 5 0 0 𝑥 3 2
  • C 1 2 5 𝑥 3 2
  • D 1 0 5 0 0 𝑥 − 3 2
  • E 8 4 𝑥 3 2

Q3:

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

Q4:

Find 𝑇 4 in the expansion of 𝑥 + 1 𝑥 1 4 .

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

Q5:

Find 𝑎 6 in the expansion of 2 4 𝑥 + 𝑦 4 7 .

  • A 6 4 𝑥 𝑦 − 2 5
  • B 1 8 9 1 6 𝑥 𝑦 5 − 2
  • C 6 4 𝑥 𝑦 5 − 2
  • D 1 8 9 1 6 𝑥 𝑦 − 2 5

Q6:

Find the third term in the expansion of 𝑎 + 𝑏 𝑎 1 9 1 1 1 4 1 9 − 2 8 .

  • A 3 7 8 𝑎 𝑏 2 6 9 1 1 7
  • B 𝑎 𝑏 8 3 1 1 7
  • C 𝑎 𝑏 2 6 9 1 1 7
  • D 3 7 8 𝑎 𝑏 8 3 1 1 7

Q7:

Consider the binomial expansion of ( 2 𝑥 − 𝑦 ) 9 in ascending powers of 𝑥 . What is the seventh term?

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

Q8:

Find the general term in 6 𝑥 − 1 6 𝑥 𝑛 + 7 .

  • A ( − 1 ) × 𝐶 × 6 × 𝑥 𝑟 𝑛 + 7 𝑟 𝑛 − 𝑟 + 7 𝑛 − 𝑟 + 7
  • B 𝑛 + 7 𝑟 𝑛 − 2 𝑟 + 7 𝑛 − 2 𝑟 + 7 𝐶 × 6 × 𝑥
  • C ( − 1 ) × 𝐶 × 6 × 𝑥 𝑟 𝑛 + 7 𝑟 + 1 𝑛 − 2 𝑟 + 7 𝑛 − 2 𝑟 + 7
  • D ( − 1 ) × 𝐶 × 6 × 𝑥 𝑟 𝑛 + 7 𝑟 𝑛 − 2 𝑟 + 7 𝑛 − 2 𝑟 + 7
  • E ( − 1 ) × 𝐶 × 6 × 𝑥 𝑟 + 1 𝑛 + 7 𝑟 𝑛 − 2 𝑟 + 7 𝑛 − 2 𝑟 + 7

Q9:

Find 𝑇 𝑛 + 3 in the expansion of 9 𝑥 − 1 𝑥 8 3 𝑛 + 9 .

  • A ( − 1 ) × 𝐶 × 9 × 𝑥 𝑛 + 2 3 𝑛 + 9 𝑛 + 2 2 𝑛 + 7 − 5 𝑛 − 2 5
  • B ( − 1 ) × 𝐶 × 9 × 𝑥 𝑛 + 3 3 𝑛 + 9 𝑛 + 3 2 𝑛 + 7 − 5 𝑛 − 1 7
  • C ( − 1 ) × 𝐶 × 9 × 𝑥 𝑛 + 3 3 𝑛 + 9 𝑛 + 3 2 𝑛 + 7 − 6 𝑛 − 9
  • D ( − 1 ) × 𝐶 × 9 × 𝑥 𝑛 + 2 3 𝑛 + 9 𝑛 + 2 2 𝑛 + 7 − 6 𝑛 − 9
  • E 3 𝑛 + 9 𝑛 + 2 2 𝑛 + 7 − 6 𝑛 − 9 𝐶 × 9 × 𝑥

Q10:

Find the value of 𝑥 that satisfies

Q11:

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

  • A 2 0 0 𝑥 2
  • B 2 0 0 0 𝑥 3
  • C 2 0 0 𝑥 3
  • D 2 0 0 0 𝑥 2

Q12:

In a binomial expansion, where the general term is 1 5 𝑟 1 8 − 9 𝑟 𝐶 𝑥 , determine the position of the term containing 𝑥 9 .

  • A 𝑇 3
  • B 𝑇 1
  • C 𝑇 4
  • D 𝑇 2

Q13:

Let 𝑇 𝑘 be the 𝑘 t h term in the expansion of ( 1 + 𝑥 ) 3 4 in increasing powers of 𝑥 . Find all nonzero values of 𝑥 for which 2 𝑇 = 𝑇 + 𝑇 2 1 2 0 2 2 .

  • A4, 12
  • B2, 14
  • C2, − 1
  • D2, 1

Q14:

Find the second-to-last term in ( 2 + 𝑥 ) 3 4 .

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

Q15:

If the ratio between the fourth term in the expansion of 𝑥 + 1 𝑥 9 and the third term in the expansion of 𝑥 − 1 𝑥 2 8 equals 7 ∶ 1 2 , find the value of 𝑥 .

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

Q16:

Consider the binomial expansion of ( 1 + 𝑥 ) 𝑛 in ascending powers of 𝑥 . Given that 𝑇 = 𝑇 8 6 when 𝑥 = 1 √ 5 , find the value of 𝑛 .

Q17:

Given that the sum of the first, middle, and last terms in the expansion of ( 𝑥 − 1 ) 6 is 42 337, find all possible real values of 𝑥 .

  • A 𝑥 = − 6 , 𝑥 = − √ 1 9 6 3
  • B 𝑥 = 6 , 𝑥 = √ 1 9 6 3
  • C 𝑥 = − 6 , 𝑥 = √ 1 9 6 3
  • D 𝑥 = 6 , 𝑥 = − √ 1 9 6 3
  • E 𝑥 = 2 1 6 , 𝑥 = − 1 9 6

Q18:

Let 𝑇 𝑘 be the 𝑘 th term in the expansion of ( 𝑥 − 2 ) 1 9 in descending powers of 𝑥 . Find all the nonzero values of 𝑥 for which 6 𝑇 − 5 𝑇 + 𝑇 = 0 8 9 1 0 .

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

Q19:

Consider the expansion of ( 1 + 𝑥 ) 𝑛 in ascending powers of 𝑥 . Given that the coefficient of 𝑥 1 4 is equal to the coefficient of 𝑇 1 8 , determine the value of 𝑛 .

Q20:

If the coefficient of the third term in the expansion of 𝑥 − 1 4 𝑛 is 3 3 8 , determine the middle term in the expansion.

  • A 2 3 1 1 0 2 4 𝑥 7
  • B 9 9 5 1 2 𝑥 6
  • C 9 9 5 1 2 𝑥 7
  • D 2 3 1 1 0 2 4 𝑥 6

Q21:

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

If 8 ( 𝑇 ) = 2 7 𝑇 × 𝑇 6 2 4 8 , what is the value of 𝑛 ?

Q22:

Consider the expansion of ( 1 + 𝑥 ) 𝑛 . Find the values of 𝑛 given that the coefficient of 𝑇 1 3 is equal to the coefficient of 𝑇 4 .

Q23:

If 𝑎 1 2 is the term free of 𝑞 in 6 𝑞 − 1 𝑞 2 𝑛 , find 𝑛 .

Q24:

Find 𝑇 3 in the expansion of 2 √ 𝑥 + √ 𝑥 2 1 3 .

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

Q25:

Find 𝑇 𝑛 − 3 in the expansion of 3 𝑥 − 1 𝑥 2 4 𝑛 + 9 .

  • A ( − 1 ) × 𝐶 × 3 × 𝑥 𝑛 − 4 4 𝑛 + 9 𝑛 − 4 3 𝑛 + 1 3 2 𝑛 + 1 7
  • B ( − 1 ) × 𝐶 × 3 × 𝑥 𝑛 − 3 4 𝑛 + 9 𝑛 − 3 3 𝑛 + 1 3 2 𝑛 + 1 9
  • C ( − 1 ) × 𝐶 × 3 × 𝑥 𝑛 − 3 4 𝑛 + 9 𝑛 − 3 3 𝑛 + 1 3 𝑛 + 2 1
  • D ( − 1 ) × 𝐶 × 3 × 𝑥 𝑛 − 4 4 𝑛 + 9 𝑛 − 4 3 𝑛 + 1 3 𝑛 + 2 1
  • E 4 𝑛 + 9 𝑛 − 4 3 𝑛 + 1 3 𝑛 + 2 1 𝐶 × 3 × 𝑥

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