Are there any positive integers that cannot be expressed as sum of at least two consecutive integers ?

Question

ganeshie8 · Answer

2 = not possible
3 = 1+2
4 = not possible
5 = 2+3
6 = 1+2+3
7 = 3+4
...

omarbirjas · Answer

8

ganeshie8 · Answer

yes 8 also cannot be expressed as sum of consecutive integers

parthkohli · Answer

Let the first number be $p$ and let there be $q$ consecutive integers.$$k = pq + \dfrac{q(q-1)}{2}$$

omarbirjas · Answer

to tired to do any more math...

parthkohli · Answer

I don't know how recasting the problem into this form helps, but let's see.

ganeshie8 · Answer

that looks good because it gives us something to play with

ganeshie8 · Answer

k = p + (p+1) + (p+2) + ... + (p+q-1)
   = pq + q(q-1)/2

so k must be of this form is it

parthkohli · Answer

exactly.

parthkohli · Answer

Umm, I haven't even answered the question...

ganeshie8 · Answer

we're 90% done actually

ganeshie8 · Answer

$$k = pq + \dfrac{q(q-1)}{2}$$
multiplying 2 through out gives
$$2k = q(2p+q-1)$$
clearly the factors on right hand side cannot have same parity (cannot be both odd or both even)

ganeshie8 · Answer

thats because the sum of factors : $q+2p+q-1 = 2(p+q)-1$ is odd

ganeshie8 · Answer

so $2k=q(2p+q-1)$ is impossible when $k$ is a power of $2$

anonymous · Answer

can someone help me with trigonometric raios? sin(3x+13) = cos(4x)