Question Video: Finding the Sum of Binomial Coefficients Mathematics • Third Year of Secondary School

Video Transcript

In the expansion of two 𝑥 plus 𝑎 all raised to the sixth power according to the descending powers of 𝑥, if the coefficient of the second term equals negative 1,344, find the value of 𝑎.

In this question, we are given a binomial expression in the form 𝑎 plus 𝑏 all raised to the 𝑛th power, where the power or exponent 𝑛 is equal to six. We can find the general expansion of this by recalling Pascal’s triangle and the binomial theorem. Pascal’s triangle is a triangular array of the binomial coefficients. As the value of 𝑛 in this case is six, our expansion will have seven terms. The first term is equal to one multiplied by 𝑎 to the sixth power. The second term is equal to six multiplied by 𝑎 to the fifth power multiplied by 𝑏 or 𝑏 to the first power. The third term is equal to 15 multiplied by 𝑎 to the fourth power multiplied by 𝑏 squared. This pattern continues as shown, where the power or exponent of 𝑎 decreases and the exponent or power of 𝑏 increases.

In this question, we need to expand two 𝑥 plus 𝑎 to the sixth power. Therefore, the value of 𝑎 from the general formula is two 𝑥 and the value of 𝑏 is 𝑎. We are interested in the second term. This will be equal to six multiplied by two 𝑥 all to the fifth power multiplied by 𝑎 to the first power or just 𝑎. As two to the fifth power is 32, two 𝑥 raised to the fifth power is 32𝑥 to the fifth power. Multiplying this by six and by 𝑎 gives us 192𝑎𝑥 to the fifth power. The coefficient here is equal to 192𝑎, and we are told in the question this is equal to negative 1,344. Clearing some space, we have 192𝑎 is equal to negative 1,344. Dividing both sides by 192 gives us a value of 𝑎 equal to negative seven.