Find the second-to-last term in two
plus 𝑥 all raised to the power of 34.
In order to answer this question,
we need to recall the binomial expansion of 𝑎 plus 𝑏 to the 𝑛th power, where 𝑛
is a natural number, as shown. The general term is equal to 𝑛
choose 𝑟 multiplied by 𝑎 to the power of 𝑛 minus 𝑟 multiplied by 𝑏 to the power
of 𝑟, where 𝑛 choose 𝑟 is equal to 𝑛 factorial divided by 𝑛 minus 𝑟 factorial
multiplied by 𝑟 factorial.
We are asked to calculate the
second-to-last term. This is equal to 𝑛 choose 𝑛 minus
one multiplied by 𝑎 to the power of one multiplied by 𝑏 to the power of 𝑛 minus
one. In this question, 𝑎 is equal to
two, 𝑏 is equal to 𝑥, and the exponent 𝑛 equals 34. This means we need to calculate 34
choose 33 multiplied by two to the power of one multiplied by 𝑥 to the power of 34
34 choose 33 is equal to 34
factorial divided by one factorial multiplied by 33 factorial. We can rewrite the numerator as 34
multiplied by 33 factorial. And as one factorial equals one,
the denominator is 33 factorial. Dividing the numerator and
denominator by 33 factorial gives us 34.
The second-to-last term simplifies
to 34 multiplied by two multiplied by 𝑥 to the power of 33. This is equal to 68𝑥 to the power