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In this lesson, we will learn how to use the rule of the ratio between two consecutive terms in a binomial expansion to find the ratio or an unknown variable.

Q1:

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

Q2:

Find the value of π given that the ratio between π 5 and π 6 , in the expansion of ( π + π ) π , equals the ratio between π 6 and π 7 in the expansion of ( π + π ) π + 9 .

Q3:

Find the ratio between the fifteenth and seventeenth terms in the expansion of ( π₯ β 1 2 ) 1 9 .

Q4:

Determine the ratio of the coefficient of π 1 3 in ( 1 + π₯ ) 8 3 to the coefficient of π 1 2 in ( 1 + π₯ ) 8 2 . Note that for π₯ > π¦ , we have π₯ π¦ π₯ β 1 π¦ β 1 πΆ πΆ = π₯ π¦ .

Q5:

Consider the expansion of οΌ 1 0 π₯ + 9 π₯ ο 2 π . Find the value of π given that π = π 1 1 1 2 and the ratio between π 1 5 and π 1 6 is equal to 5 βΆ 3 .

Q6:

The coefficients of three consecutive terms in the expansion of ( 1 + π₯ ) π are 230, 690, and 1β380 respectively. Evaluate π and find their orders.

Q7:

Consider the expansion of οΏ β π₯ + 1 1 β π₯ ο 1 3 . Find π₯ given that the ratio between π 1 3 and π 1 4 is 3 1 βΆ 1 5 .

Q8:

Consider the expansion of οΌ 5 2 + 2 π₯ 5 ο π 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 βΆ 3 7 5 , determine the value of π .

Q9:

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

Q10:

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

Q11:

Consider the expansion of If the ratio between the middle term and the term containing π₯ β 1 1 is 2 0 βΆ 2 1 , determine the value of π₯ .

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.

Q13:

Note that π π π β 1 π β 1 πΆ βΆ πΆ = π βΆ π . Given that the ratio between the coefficient of π 2 7 in the binomial expansion of οΉ 1 + π₯ ο 2 π and the coefficient of π 2 6 in the expansion of ( 1 β π¦ ) π β 1 is β 2 9 βΆ 1 3 , find the value of π .

Q14:

Find π₯ given that the ratio between the sixth and the seventh terms in the expansion of οΌ 1 2 π₯ + 6 π₯ ο 2 1 0 is equal to 7 5 βΆ 2 .

Q15:

Find π₯ given that the ratio between the second and the third terms in the expansion of οΌ 1 5 π₯ + 2 π₯ ο 2 1 1 is equal to 1 2 5 βΆ 1 8 .

Q16:

Find π₯ given that the ratio between the ninth and the tenth terms in the expansion of οΌ 1 5 π₯ + 8 π₯ ο 2 1 2 is equal to 8 βΆ 9 .

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 βΆ 2 , find the value of π₯ .

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