Answer to Question #323500 in Algebra for ild

Question

1) Without fully expanding (4a-b)7, determine the fifth term.

2) Find the middle term of (2c - 12𝑑)10

Answer 1

$1:\\\left( 4a-b \right) ^7\\The\,\,fifth\,\,term\,\,\left( counting\,\,from\,\,0th\,\,term \right) \,\,is\,\,\\C_{7}^{5}\left( 4a \right) ^5\left( -b \right) ^{7-5}=21504a^5b^2\\2:\\\left( 2c-12d \right) ^{10}\\The\,\,middle\,\,term\,\,is\,\,for\,\,5th\,\,powers:\\C_{10}^{5}\left( 2c \right) ^5\left( -12d \right) ^5=-2006581248c^5d^5$