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Lesson: Ratio between Two Consecutive Terms in a Binomial

Sample Question Videos

Worksheet • 17 Questions • 1 Video

Q1:

Consider the expansion of ( 8 π‘₯ + 2 𝑦 ) 2 3 . Find the ratio between the eighth and the seventh terms.

  • A 1 7 𝑦 2 8 π‘₯
  • B 8 π‘₯ 𝑦
  • C 2 8 𝑦 1 7 π‘₯
  • D 2 8 π‘₯ 1 7 𝑦
  • E 1 7 π‘₯ 2 8 𝑦

Q2:

Find the value of 𝑛 given that the ratio between 𝑇 5 and 𝑇 6 , in the expansion of ( π‘Ž + 𝑏 ) 𝑛 , equals the ratio between 𝑇 6 and 𝑇 7 in the expansion of ( π‘Ž + 𝑏 ) 𝑛 + 9 .

Q3:

Find the ratio between the fifteenth and seventeenth terms in the expansion of ( π‘₯ βˆ’ 1 2 ) 1 9 .

  • A π‘₯ 1 2 2
  • B π‘₯ 1 2
  • C 1 2 π‘₯
  • D 1 2 π‘₯ 2

Q4:

Determine the ratio of the coefficient of 𝑇 1 3 in ( 1 + π‘₯ ) 8 3 to the coefficient of 𝑇 1 2 in ( 1 + π‘₯ ) 8 2 . Note that for π‘₯ > 𝑦 , we have π‘₯ 𝑦 π‘₯ βˆ’ 1 𝑦 βˆ’ 1 𝐢 𝐢 = π‘₯ 𝑦 .

  • A 8 3 ∢ 1 2
  • B 1 2 ∢ 8 3
  • C 1 3 ∢ 8 3
  • D 8 3 ∢ 1 3

Q5:

Consider the expansion of ο€Ό 1 0 π‘₯ + 9 π‘₯  2 𝑛 . Find the value of 𝑛 given that 𝑇 = 𝑇 1 1 1 2 and the ratio between 𝑇 1 5 and 𝑇 1 6 is equal to 5 ∢ 3 .

Q6:

The coefficients of three consecutive terms in the expansion of ( 1 + π‘₯ ) 𝑛 are 230, 690, and 1 380 respectively. Evaluate 𝑛 and find their orders.

  • A11, 𝑇 3 , 𝑇 4 , 𝑇 5
  • B11, 𝑇 1 2 , 𝑇 1 3 , 𝑇 1 4
  • C11, 𝑇 4 , 𝑇 5 , 𝑇 6
  • D11, 𝑇 1 1 , 𝑇 1 2 , 𝑇 1 3

Q7:

Consider the expansion of ο€Ώ √ π‘₯ + 1 1 √ π‘₯  1 3 . Find π‘₯ given that the ratio between 𝑇 1 3 and 𝑇 1 4 is 3 1 ∢ 1 5 .

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

Q8:

Consider the expansion of ο€Ό 5 2 + 2 π‘₯ 5  𝑛 in ascending powers of π‘₯ . Given that the ratio between the coefficient of the fourth term and the coefficient of the second term is 1 7 6 ∢ 3 7 5 , determine the value of 𝑛 .

Q9:

Consider the expansion of ( π‘š π‘₯ + 8 ) 𝑛 . Determine the values of π‘š and 𝑛 , given that the ratio between the coefficients of 𝑇 1 2 and 𝑇 1 4 is equal to 6 3 7 4 6 4 0 and that the ratio between the coefficients of 𝑇 7 and 𝑇 9 is equal to 4 9 1 3 6 0 .

  • A 𝑛 = 4 1 , π‘š = 7
  • B 𝑛 = 7 , π‘š = 4 1
  • C 𝑛 = 7 , π‘š = 3 4
  • D 𝑛 = 3 4 , π‘š = 7

Q10:

The ratio between the coefficients of three consecutive terms in the expansion of ( 1 + π‘₯ ) 𝑛 is 2 2 ∢ 2 3 ∢ 2 2 . Evaluate 𝑛 and find the orders of these terms.

  • A44, 𝑇 2 2 , 𝑇 2 3 , 𝑇 2 4
  • B44, 𝑇 2 3 , 𝑇 2 4 , 𝑇 2 5
  • C22, 𝑇 2 3 , 𝑇 2 4 , 𝑇 2 5
  • D22, 𝑇 2 2 , 𝑇 2 3 , 𝑇 2 4

Q11:

Consider the expansion of If the ratio between the middle term and the term containing π‘₯ βˆ’ 1 1 is 2 0 ∢ 2 1 , determine the value of π‘₯ .

  • A2
  • B 1 2
  • C 1 4
  • D4

Q12:

Consider the binomial expansion of ( 3 + 7 π‘₯ ) 2 8 in ascending powers of π‘₯ . When π‘₯ = 6 , one of the terms in the expansion is equal to twice its following term. Find the position of these two terms.

  • A 𝑇 2 8 , 𝑇 2 9
  • B 𝑇 2 8 , 𝑇 2 7
  • C 𝑇 2 7 , 𝑇 2 6
  • D 𝑇 2 9 , 𝑇 2 7

Q13:

Note that 𝑛 π‘Ÿ 𝑛 βˆ’ 1 π‘Ÿ βˆ’ 1 𝐢 ∢ 𝐢 = 𝑛 ∢ π‘Ÿ . Given that the ratio between the coefficient of 𝑇 2 7 in the binomial expansion of ο€Ή 1 + π‘₯  2 𝑛 and the coefficient of 𝑇 2 6 in the expansion of ( 1 βˆ’ 𝑦 ) 𝑛 βˆ’ 1 is βˆ’ 2 9 ∢ 1 3 , find the value of 𝑛 .

Q14:

Find π‘₯ given that the ratio between the sixth and the seventh terms in the expansion of ο€Ό 1 2 π‘₯ + 6 π‘₯  2 1 0 is equal to 7 5 ∢ 2 .

  • A 5 2
  • B 2 5
  • C 8 1 2 5
  • D 1 2 5 8

Q15:

Find π‘₯ given that the ratio between the second and the third terms in the expansion of ο€Ό 1 5 π‘₯ + 2 π‘₯  2 1 1 is equal to 1 2 5 ∢ 1 8 .

  • A 5 3
  • B 3 5
  • C 2 7 1 2 5
  • D 1 2 5 2 7

Q16:

Find π‘₯ given that the ratio between the ninth and the tenth terms in the expansion of ο€Ό 1 5 π‘₯ + 8 π‘₯  2 1 2 is equal to 8 ∢ 9 .

  • A 2 3
  • B 3 2
  • C 2 7 8
  • D 8 2 7

Q17:

Consider the expansion of ( 3 + 6 π‘₯ ) 1 3 in ascending powers of π‘₯ . Given that the ratio between the thirteenth term and the twelfth term is 3 ∢ 2 , find the value of π‘₯ .

  • A 9 2
  • B 2 9
  • C2
  • D 1 2
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