Binomial expansion/pascal's triangle. with all of number 5.

The rows add up to , respectively. (Notice they're all powers of 2)

The sum of the numbers in row is .

The last problem can be solved with the binomial theorem, but I'll assume you don't take that for granted. You can prove this claim by induction. When ,



so the base case holds. Assume the claim holds for , so that



Use this to show that it holds for .





Notice that








So you can write the expansion for as



and since , you have



and so the claim holds for , thus proving the claim overall that



Setting gives



which agrees with the result obtained for part (c).

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