The solution of equation $4+6+8+10+\cdots +x =270$ is 15. The solution of equation 
$4+6+8+10+\cdots +x =270$ 
is $x=15$. 
How can I prove it? 
I ve tried the geometric sequence but I cannot figure out the pattern.

Since $$1+2+3+4+.....+(n-1)+n = \frac{n(n+1)}{2}$$
$$2+4+6+8+.....+2(n-1)+2n = n(n+1)$$
That is $$4+6+8+.....+2(n-1)+2n = n(n+1)-2$$
Thus $$n(n+1)-2 \leq 270$$
$$n^2+n-272 \leq 0$$
$$(n+17)(n-16) \leq 0$$
So $$n=16$$
Thus $$4+6+8+10+12+14+16+18+20+22+24+26+28+30+32=270$$
I think the question should be $$(4+6+8+...)\cdot x =270$$
Then we can argue that factors of $270$ are $1,2,3,5,6,9,10,15,18,27,30,45,54,90,135,270$
Then the only chances are 
$$4+6 =10 $$ which gives $x=27$
$$4+6+8 = 18$$ which gives $x=15$
$$4+6+8+10+12+14 =54 $$ which gives $x=5$
$$4+6+8+10+12+14+16+18+20+22+24+26+28+30+32 = 270 $$ which gives $x=1$
