Worksheet: General Term in the Binomial Theorem

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 ๐‘Ž๏Šช in the expansion of ๏€ฟ5โˆš๐‘ฅ+โˆš๐‘ฅ5๏‹๏Šฏ.

  • A10,500๐‘ฅ๏Žข๏Žก
  • B125๐‘ฅ๏Žข๏Žก
  • C84๐‘ฅ๏Žข๏Žก
  • D10,500๐‘ฅ๏Šฑ๏Žข๏Žก
  • E125๐‘ฅ๏Šฑ๏Žข๏Žก

Q2:

Find ๐‘Ž๏Šช in the expansion of ๏€ผ๐‘ฅ+1๐‘ฅ๏ˆ๏Šง๏Šช.

  • A๐‘ฅ๏Šช
  • B๐‘ฅ๏Šฎ
  • C364๐‘ฅ๏Šฎ
  • D728๐‘ฅ๏Šฎ

Q3:

Find ๐‘Ž๏Šฌ in the expansion of ๏€ผ24๐‘ฅ+๐‘ฆ4๏ˆ๏Šญ.

  • A64๐‘ฅ๐‘ฆ๏Šซ๏Šฑ๏Šจ
  • B18916๐‘ฅ๐‘ฆ๏Šซ๏Šฑ๏Šจ
  • C64๐‘ฅ๐‘ฆ๏Šฑ๏Šจ๏Šซ
  • D18916๐‘ฅ๐‘ฆ๏Šฑ๏Šจ๏Šซ

Q4:

Find the third term in the expansion of ๏€ฝ๐‘Ž+๐‘๐‘Ž๏‰๏Ž ๏Žจ๏Ž ๏Ž ๏Ž ๏Žฃ๏Ž ๏Žจ๏Šฑ๏Šจ๏Šฎ.

  • A๐‘Ž๐‘๏Žก๏Žฅ๏Žจ๏Ž ๏Ž ๏Žฆ
  • B๐‘Ž๐‘๏Žง๏Žข๏Ž ๏Ž ๏Žฆ
  • C378๐‘Ž๐‘๏Žง๏Žข๏Ž ๏Ž ๏Žฆ
  • D378๐‘Ž๐‘๏Žก๏Žฅ๏Žจ๏Ž ๏Ž ๏Žฆ

Q5:

Consider the binomial expansion of (2๐‘ฅโˆ’๐‘ฆ)๏Šฏ in ascending powers of ๐‘ฅ. What is the seventh term?

  • A5,376๐‘ฅ๐‘ฆ๏Šฌ๏Šฉ
  • Bโˆ’672๐‘ฅ๐‘ฆ๏Šฉ๏Šฌ
  • Cโˆ’5,376๐‘ฅ๐‘ฆ๏Šฌ๏Šฉ
  • D672๐‘ฅ๐‘ฆ๏Šฉ๏Šฌ

Q6:

Find the general term in ๏€ผ6๐‘ฅโˆ’16๐‘ฅ๏ˆ๏Š๏Šฐ๏Šญ.

  • A๏Š๏Šฐ๏Šญ๏Ž๏Š๏Šฑ๏Šจ๏Ž๏Šฐ๏Šญ๏Š๏Šฑ๏Šจ๏Ž๏Šฐ๏Šญ๐ถร—6ร—๐‘ฅ
  • B(โˆ’1)ร—๐ถร—6ร—๐‘ฅ๏Ž๏Š๏Šฐ๏Šญ๏Ž๏Š๏Šฑ๏Šจ๏Ž๏Šฐ๏Šญ๏Š๏Šฑ๏Šจ๏Ž๏Šฐ๏Šญ
  • C(โˆ’1)ร—๐ถร—6ร—๐‘ฅ๏Ž๏Š๏Šฐ๏Šญ๏Ž๏Š๏Šฑ๏Ž๏Šฐ๏Šญ๏Š๏Šฑ๏Ž๏Šฐ๏Šญ
  • D(โˆ’1)ร—๐ถร—6ร—๐‘ฅ๏Ž๏Š๏Šฐ๏Šญ๏Ž๏Šฐ๏Šง๏Š๏Šฑ๏Šจ๏Ž๏Šฐ๏Šญ๏Š๏Šฑ๏Šจ๏Ž๏Šฐ๏Šญ
  • E(โˆ’1)ร—๐ถร—6ร—๐‘ฅ๏Ž๏Šฐ๏Šง๏Š๏Šฐ๏Šญ๏Ž๏Š๏Šฑ๏Šจ๏Ž๏Šฐ๏Šญ๏Š๏Šฑ๏Šจ๏Ž๏Šฐ๏Šญ

Q7:

Find ๐‘Ž๏Š๏Šฐ๏Šฉ in the expansion of ๏€ผ9๐‘ฅโˆ’1๐‘ฅ๏ˆ๏Šฎ๏Šฉ๏Š๏Šฐ๏Šฏ.

  • A๏Šฉ๏Š๏Šฐ๏Šฏ๏Š๏Šฐ๏Šจ๏Šจ๏Š๏Šฐ๏Šญ๏Šฑ๏Šฌ๏Š๏Šฑ๏Šฏ๐ถร—9ร—๐‘ฅ
  • B(โˆ’1)ร—๐ถร—9ร—๐‘ฅ๏Š๏Šฐ๏Šฉ๏Šฉ๏Š๏Šฐ๏Šฏ๏Š๏Šฐ๏Šฉ๏Šจ๏Š๏Šฐ๏Šญ๏Šฑ๏Šฌ๏Š๏Šฑ๏Šฏ
  • C(โˆ’1)ร—๐ถร—9ร—๐‘ฅ๏Š๏Šฐ๏Šฉ๏Šฉ๏Š๏Šฐ๏Šฏ๏Š๏Šฐ๏Šฉ๏Šจ๏Š๏Šฐ๏Šญ๏Šฑ๏Šซ๏Š๏Šฑ๏Šง๏Šญ
  • D(โˆ’1)ร—๐ถร—9ร—๐‘ฅ๏Š๏Šฐ๏Šจ๏Šฉ๏Š๏Šฐ๏Šฏ๏Š๏Šฐ๏Šจ๏Šจ๏Š๏Šฐ๏Šญ๏Šฑ๏Šฌ๏Š๏Šฑ๏Šฏ
  • E(โˆ’1)ร—๐ถร—9ร—๐‘ฅ๏Š๏Šฐ๏Šจ๏Šฉ๏Š๏Šฐ๏Šฏ๏Š๏Šฐ๏Šจ๏Šจ๏Š๏Šฐ๏Šญ๏Šฑ๏Šซ๏Š๏Šฑ๏Šจ๏Šซ

Q8:

Find the third term in the expansion of ๏€ฟ2๐‘ฅ+5โˆš๐‘ฅ๏‹๏Šซ.

  • A2,000๐‘ฅ๏Šฉ
  • B200๐‘ฅ๏Šฉ
  • C2,000๐‘ฅ๏Šจ
  • D200๐‘ฅ๏Šจ

Q9:

In a binomial expansion, where the general term is ๏Šง๏Šซ๏Ž๏Šง๏Šฎ๏Šฑ๏Šฏ๏Ž๐ถ๐‘ฅ, determine the position of the term containing ๐‘ฅ๏Šฏ.

  • A๐‘Ž๏Šช
  • B๐‘Ž๏Šจ
  • C๐‘Ž๏Šง
  • D๐‘Ž๏Šฉ

Q10:

Let ๐‘Ž๏‡ be the ๐‘˜th term in the expansion of (1+๐‘ฅ)๏Šฉ๏Šช in increasing powers of ๐‘ฅ. Find all nonzero values of ๐‘ฅ for which 2๐‘Ž=๐‘Ž+๐‘Ž๏Šจ๏Šง๏Šจ๏Šฆ๏Šจ๏Šจ.

  • A2, 1
  • B4, 12
  • C2, โˆ’1
  • D2, 14

Q11:

Find the second-to-last term in (2+๐‘ฅ)๏Šฉ๏Šช.

  • A34๐‘ฅ๏Šฉ๏Šฉ
  • B68๐‘ฅ
  • C68๐‘ฅ๏Šฉ๏Šฉ
  • D34๐‘ฅ

Q12:

If the ratio between the fourth term in the expansion of ๏€ผ๐‘ฅ+1๐‘ฅ๏ˆ๏Šฏ and the third term in the expansion of ๏€ผ๐‘ฅโˆ’1๐‘ฅ๏ˆ๏Šจ๏Šฎ equals 7โˆถ12, find the value of ๐‘ฅ.

  • A1,29649
  • B491,296
  • C736
  • D367

Q13:

Consider the binomial expansion of (1+๐‘ฅ)๏Š in ascending powers of ๐‘ฅ. Given that ๐‘Ž=๐‘Ž๏Šฎ๏Šฌ when ๐‘ฅ=1โˆš5, find the value of ๐‘›.

Q14:

Given that the sum of the first, middle, and last terms in the expansion of (๐‘ฅโˆ’1)๏Šฌ is 42,337, find all possible real values of ๐‘ฅ.

  • A๐‘ฅ=6, ๐‘ฅ=โˆš196๏Žข
  • B๐‘ฅ=โˆ’6, ๐‘ฅ=โˆš196๏Žข
  • C๐‘ฅ=โˆ’6, ๐‘ฅ=โˆ’โˆš196๏Žข
  • D๐‘ฅ=6, ๐‘ฅ=โˆ’โˆš196๏Žข
  • E๐‘ฅ=216, ๐‘ฅ=โˆ’196

Q15:

Let ๐‘Ž๏‡ be the ๐‘˜th term in the expansion of (๐‘ฅโˆ’2)๏Šง๏Šฏ in descending powers of ๐‘ฅ. Find all the nonzero values of ๐‘ฅ for which 6๐‘Žโˆ’5๐‘Ž+๐‘Ž=0๏Šฎ๏Šฏ๏Šง๏Šฆ.

  • A๐‘ฅ=โˆ’43, ๐‘ฅ=โˆ’113
  • B๐‘ฅ=โˆ’43, ๐‘ฅ=23
  • C๐‘ฅ=โˆ’113, ๐‘ฅ=23
  • D๐‘ฅ=โˆ’23, ๐‘ฅ=โˆ’116

Q16:

Consider the expansion of (1+๐‘ฅ)๏Š in ascending powers of ๐‘ฅ. Given that the coefficient of ๐‘ฅ๏Šง๏Šช is equal to the coefficient of ๐‘Ž๏Šง๏Šฎ, determine the value of ๐‘›.

Q17:

If the coefficient of the third term in the expansion of ๏€ผ๐‘ฅโˆ’14๏ˆ๏Š is 338, determine the middle term in the expansion.

  • A99512๐‘ฅ๏Šญ
  • B99512๐‘ฅ๏Šฌ
  • C2311,024๐‘ฅ๏Šญ
  • D2311,024๐‘ฅ๏Šฌ

Q18:

In the binomial expansion of (1+๐‘ฅ)๏Š, ๐‘› is a positive, whole number and ๐‘Ž๏Ž is the ๐‘Ÿth term, or the term which contains ๐‘ฅ๏Ž๏Šฑ๏Šง.

If 8(๐‘Ž)=27๐‘Žร—๐‘Ž๏Šฌ๏Šจ๏Šช๏Šฎ, what is the value of ๐‘›?

Q19:

Consider the expansion of (1+๐‘ฅ)๏Š. Find the values of ๐‘› given that the coefficient of ๐‘Ž๏Šง๏Šฉ is equal to the coefficient of ๐‘Ž๏Šช.

Q20:

If ๐‘Ž๏Šง๏Šจ is the term free of ๐‘ž in ๏€ฝ6๐‘žโˆ’1๐‘ž๏‰๏Šจ๏Š, find ๐‘›.

Q21:

Find ๐‘Ž๏Šฉ in the expansion of ๏€ฟ2โˆš๐‘ฅ+โˆš๐‘ฅ2๏‹๏Šง๏Šฉ.

  • A39,936๐‘ฅ๏Žจ๏Žก
  • B512๐‘ฅ๏Žจ๏Žก
  • C78๐‘ฅ๏Žจ๏Žก
  • D39,936๐‘ฅ๏Šฑ๏Žจ๏Žก
  • E512๐‘ฅ๏Šฑ๏Žจ๏Žก

Q22:

Find ๐‘Ž๏Š๏Šฑ๏Šฉ in the expansion of ๏€ผ3๐‘ฅโˆ’1๐‘ฅ๏ˆ๏Šจ๏Šช๏Š๏Šฐ๏Šฏ.

  • A๏Šช๏Š๏Šฐ๏Šฏ๏Š๏Šฑ๏Šช๏Šฉ๏Š๏Šฐ๏Šง๏Šฉ๏Š๏Šฐ๏Šจ๏Šง๐ถร—3ร—๐‘ฅ
  • B(โˆ’1)ร—๐ถร—3ร—๐‘ฅ๏Š๏Šฑ๏Šฉ๏Šช๏Š๏Šฐ๏Šฏ๏Š๏Šฑ๏Šฉ๏Šฉ๏Š๏Šฐ๏Šง๏Šฉ๏Š๏Šฐ๏Šจ๏Šง
  • C(โˆ’1)ร—๐ถร—3ร—๐‘ฅ๏Š๏Šฑ๏Šฉ๏Šช๏Š๏Šฐ๏Šฏ๏Š๏Šฑ๏Šฉ๏Šฉ๏Š๏Šฐ๏Šง๏Šฉ๏Šจ๏Š๏Šฐ๏Šง๏Šฏ
  • D(โˆ’1)ร—๐ถร—3ร—๐‘ฅ๏Š๏Šฑ๏Šช๏Šช๏Š๏Šฐ๏Šฏ๏Š๏Šฑ๏Šช๏Šฉ๏Š๏Šฐ๏Šง๏Šฉ๏Š๏Šฐ๏Šจ๏Šง
  • E(โˆ’1)ร—๐ถร—3ร—๐‘ฅ๏Š๏Šฑ๏Šช๏Šช๏Š๏Šฐ๏Šฏ๏Š๏Šฑ๏Šช๏Šฉ๏Š๏Šฐ๏Šง๏Šฉ๏Šจ๏Š๏Šฐ๏Šง๏Šญ

Q23:

Find the coefficient of ๐‘ฅ๏Šง๏Šฆ in the expansion of ๏€น1+๐‘ฅโˆ’๐‘ฅ๏…๏Šจ๏Šฎ.

Q24:

Answer the following questions for the expansion of (2+4๐‘ฅ)๏Š.

Given that the coefficient of ๐‘ฅ๏Šจ is 3,840, find ๐‘›.

  • A๐‘›=9
  • B๐‘›=7
  • C๐‘›=6
  • D๐‘›=8
  • E๐‘›=5

Hence, work out the value of the coefficient of ๐‘ฅ๏Šซ.

Q25:

For which values of ๐‘ฅ are the two middle terms of ๏€พ16๐‘ฅ+๐‘ฅ625๏Š๏Šฉ๏Šง๏Šฏ equal?

  • A100,โˆ’100
  • B10,โˆ’10
  • C50,โˆ’50
  • D25,โˆ’25

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