Given-a-set-S-a-1-a-n-a-k-Z-a-1-a-a-n-a-n-1-1-a-n-b-S-contains-all-integers-between-a-and-b-Are-there-more-even-or-odd-values-

Question Number 3055 by Filup last updated on 04/Dec/15

Given a set S = {a_{1}, \dots, a_{n}}

a_{k} \in Z

a_{1} = a, a_{n} = a_{n - 1} + 1, a_{n} = b

∴ S contains all integers between

a and b .

Are there more even or odd values ?

Answered by 123456 last updated on 04/Dec/15

a_{1} = a

a_{n} = a_{n - 1} + 1

a_{n} = b (b ⩾ a)

a_{1} = a

a_{2} = a + 1

⋮

a_{n} = b = a + (b - a)

n = b - a + 1

if n = b - a + 1 \equiv 0 (\mod 2) \Rightarrow b - a \equiv 1 (\mod 2)

a_{1} \equiv a_{3} \equiv a_{5} \equiv \dots \equiv a_{n - 1}

a_{2} \equiv a_{4} \equiv a_{6} \equiv \dots \equiv a_{n}

so there same quantity of even and odd

if n = b - a + 1 \equiv 1 (\mod 2) \Rightarrow b - a \equiv 0 (\mod 2)

a_{1} \equiv a_{3} \equiv \dots \equiv a_{n - 2} \equiv a_{n}

a_{2} \equiv a_{4} \equiv \dots \equiv a_{n - 1}

so its not have same quantinty, also

a_{1} determines wich will have more

wich mean

there more numbers of parity of a_{1}

Answered by Rasheed Soomro last updated on 04/Dec/15

C a s e - 1 : a, b \in E

M o r e e v e n n u m b e r s

C a s e - 2 : a, b \in O

M o r e o d d n u m b e r s

C a s e - 3 : O n e o f a a n d b i s o d d o t h e r i s e v e n

B o t h a r e e q u a l