Video Transcript
Find the sum of the coefficients of the first three terms that result from the expansion of 𝑥 plus two all to the fifth power according to the descending powers of 𝑥.
In this question, we are given a binomial expansion of the form 𝑎 plus 𝑏 all raised to the 𝑛th power, where the value of 𝑛 is equal to five. We will use Pascal’s triangle and the binomial theorem to recall how we can expand an expression of this type.
Pascal’s triangle is a triangular array of the binomial coefficients. In this case, we are given that 𝑛 is equal to five. This means that 𝑎 plus 𝑏 all raised to the fifth power will have six terms. The first term is equal to one multiplied by 𝑎 to the fifth power. The second term is equal to five multiplied by 𝑎 to the fourth power multiplied by 𝑏 or 𝑏 to the first power. The next term is equal to 10 multiplied by 𝑎 cubed multiplied by 𝑏 squared. This pattern continues as shown, where the powers or exponents of 𝑎 decrease and the powers or exponents of 𝑏 increase.
In this question, we are told that 𝑎 is equal to 𝑥 and 𝑏 is equal to two. We are also only interested in the first three terms. 𝑥 plus two all raised to the fifth power is therefore equal to one multiplied by 𝑥 to the fifth power plus five 𝑥 to the fourth power multiplied by two to the power of one plus 10𝑥 cubed multiplied by two squared. This simplifies to one 𝑥 to the fifth power plus 10𝑥 to the fourth power plus 40𝑥 cubed, and so on. We are interested in the coefficients of these first three terms. The coefficients are equal to one, 10, and 40. In order to calculate their sum, we simply add one, 10, and 40. This is equal to 51. The sum of the coefficients of the first three terms from the expansion 𝑥 plus two all raised to the fifth power is 51.
