Question Video: Finding the Ratio between Two Terms in a Binomial Expansion Mathematics

Find the ratio between the fifteenth and seventeenth terms in the expansion of (π‘₯ βˆ’ 12)¹⁹.

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Video Transcript

Find the ratio between the 15th and 17th terms in the expansion of π‘₯ minus 12 to the 19th power.

In order to answer this question, we will use two formulas linked to the expansion of a binomial expression of the form π‘Ž plus 𝑏 to the 𝑛th power. We know that the general term of this expansion, π‘Ž sub π‘Ÿ plus one, is equal to 𝑛 choose π‘Ÿ multiplied by π‘Ž to the power of 𝑛 minus π‘Ÿ multiplied by 𝑏 to the power of π‘Ÿ. We also know that the ratio of consecutive terms of a binomial expansion, π‘Ž sub π‘Ÿ plus one over π‘Ž sub π‘Ÿ, is equal to 𝑛 minus π‘Ÿ plus one over π‘Ÿ multiplied by 𝑏 over π‘Ž.

In this question, we are dealing with the 15th and 17th terms in the expansion. The 15th term is equal to 19 choose 14 multiplied by π‘₯ to the fifth power multiplied by negative 12 to the 14th power. The 17th term is equal to 19 choose 16 multiplied by π‘₯ cubed multiplied by negative 12 to the 16th power. We can divide π‘₯ to the fifth power and π‘₯ cubed by π‘₯ cubed, leaving us with π‘₯ squared on the numerator. Likewise, dividing the numerator and denominator by negative 12 to the 14th power leaves us with negative 12 squared on the denominator. This is the same as π‘Ž squared over 𝑏 squared.

We now need to consider what happens when we divide nonconsecutive combinations. As the 17th term is two terms after the 15th term, we have π‘Ž sub π‘Ÿ over π‘Ž sub π‘Ÿ plus two. The combinations part of this will be equal to 𝑛 choose π‘Ÿ minus one over 𝑛 choose π‘Ÿ plus one. When writing this in terms of factorials, it looks quite complicated. However, we will quite quickly see that some terms cancel. We can then use our properties of factorials and the knowledge that dividing a fraction by another fraction is the same as multiplying the first fraction by the reciprocal of the second. The combinations part is therefore equal to π‘Ÿ multiplied by π‘Ÿ plus one divided by 𝑛 minus π‘Ÿ plus one multiplied by 𝑛 minus π‘Ÿ.

We can now see the link between this and the ratio of consecutive terms. We can now go back to our question to calculate 19 choose 14 divided by 19 choose 16. As 𝑛 is equal to 19 and π‘Ÿ is equal to 15, we have 15 multiplied by 16 divided by five multiplied by four. We know that negative 12 squared is 144. So we need to multiply the first part by π‘₯ squared over 144. This in turn simplifies to 12π‘₯ squared over 144. Finally, we can divide the numerator and denominator by 12 so that the ratio between the 15th and 17th terms is π‘₯ squared over 12.

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