Question

how can find
sum (x) from x=1 to x=n

Answer 1

That is just 1+2+...+n

I'm sure you have seen that before.

Answer 2

prove sum(x)=n(n+1)/2 from x=1 to x=n

Answer 3

time to review proof by induction.

Assume the sum is true for k=n.
Show that if it is, then it is also true for k=n+1

1 = 1(2)/2

So, it is clear that the equation is true for k=1.

Now, assume it is true for k=n. That is,

1+2+...+n = n(n+1)/2

Now, add n+1 to both sides:

1+2+...+n+n+1 = n(n+1)/2 + n+1
= [n(n+1) + 2(n+1)]/2
= (n+1)(n+2)/2

So, if it is true for k=n, it is also true for k=n+1.
It is true for k=1, so it then follows that it is true for 2,3,4,... and in fact, for all n.

Answer 4

thank you Steve