If n+1 integers are chosen from the set 1,2,3,…,2n
,where n is a positive integer, must at least one of them be even? Why?

Pigeonhole Principle

Step-by-step explanation:

From series {1,22n} ; we need to select 'n + 1' integers.

Even integers series from above series = {2,4,62(n-1), 2n}

Odd integers series from above series = {1,3,52(n-2), 2n-1}

So, we can choose only n odd integers from the series. Hence, if we choose n+1 integers, atleast one of them have to be even integer.

Eg : n = 5, Series = (1,2,3,4,5,6,7,8,9,10}, Odd series = {1,3,5,7,9}, Even series = {2,4,6,8,10}. If we choose 5 +1 ie 6 integers, there must be one even integer, as there are only 5 total odd integers in series.

Mike has a choice of an account that compounds daily and an account that compounds monthly. If they both have the same interest rate, which will earn more interest?

Account that is being compounded on a daily basis will earn more interest. This is because in simple words, one is getting some interest on a daily basis and one will yield an 'interest on interest'. Whereas in a monthly compounded account Mike will earn less interest.