i have a homework with 10 question but im stuck with 3 i searched about them everywhere read other c

Derick Richard 2022-05-07 Answered
i have a homework with 10 question but im stuck with 3 i searched about them everywhere read other colleges lectures but i couldnt solved them finally i desired to ask here
Question-2:
Prove that each positive integer can be written in form of 2 k q, where q is odd, and k is a non-negative integer.
Hint: Use induction, and the fact that the product of two odd numbers is odd.
Question-6:
( x + y ) n = k = 0 n C ( n , k ) x n k y k = n 2 + n 2
Prove the above statement by using induction on n.
Question-7:
Let n 1 , n 2 , . . . , n t be positive integers. Show that if n 1 + n 2 + . . . + n t t + 1 objects are placed into t boxes, then for some i, i = 1 , 2 , 3 , . . . , t , the ith box contains at least n i objects.
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Answers (2)

Kylan Simon
Answered 2022-05-08 Author has 17 answers
(2) Certainly we can write 1 = 2 0 1. Now suppose that n > 1, and every positive integer less than n can be written in the desired form. (This is your induction hypothesis.) Then either n is odd, or n is even. If n is odd, we can write n = 2 0 n to express n in the desired form. If n is even, then n = 2 m for some positive integer m < n. By the induction hypothesis there are a non-negative integer k and an odd integer q such that m = 2 k q; how can you use this to express n in the desired form? (This is not quite the proof that the author of the hint had in mind; it’s actually a bit easier.)

(6) As stated, this does not make sense. It’s true that
( x + y ) n = k = 0 n ( n k ) x n k y k ;
this is the binomial theorem. It is not true that this equals n 2 + n 2 ; this is obvious just from the fact that n 2 + n 2 doesn’t depend on x and y. What is true is that
k = 0 n k = n 2 + n 2 ;
is this what you’re actually supposed to prove?
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agrejas0hxpx
Answered 2022-05-09 Author has 4 answers
(7) Let m i be the number of objects in the i-th box, and suppose that m i < n i for i = 1 , , t. Then m i n i 1 for i = 1 , , t, so how big can i = 1 t m i be?
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