Question

for some integer q,every odd integer is of the form

Answer 1

We know that, odd intergers are 1,3,5,...

So, it can be written in the form of 2q + 1.

where, q = integer = z

or q = ...,-1,0,1,2,3,...

∴ 2q + 1 = ...,-3,-1,1,3,5,...

Alternate method

Let 'a' be the given positive integer .On dividing 'a' by 2, let q be the quotient and r be the remainder. Then, by Euclid's division algorithm, we have

a = 2q + r, where 0 ≤ r < 2

⇒ a = 2q + r, where r = 0 or r = 1

⇒ a = 2q or 2q + 1

when a = 2q + 1 for some integer q, then clearly a is odd.

Answer 2

ANSWER :

2Q+1

EXPLANATION :

We know that, odd integers are 1,3,5,...

So, it can be written in the form of 2q + 1.

where, q = integer = z

or q = ...,-1,0,1,2,3,...

∴ 2q + 1 = ...,-3,-1,1,3,5,...