how can find
sum (x) from x=1 to x=n
That is just 1+2+...+n
I'm sure you have seen that before.
prove sum(x)=n(n+1)/2 from x=1 to x=n
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.
thank you Steve
To find the sum of a series from x=1 to x=n, you can use the formula for the sum of an arithmetic series:
Sum = (n/2) * (first term + last term)
In this case, since the first term is 1 and the last term is n, the formula becomes:
Sum = (n/2) * (1 + n)
Now, let's apply this formula to an example. Let's say we want to find the sum of the series from x=1 to x=5:
n = 5
Sum = (5/2) * (1 + 5)
= (5/2) * 6
= 15
So, the sum of the series from x=1 to x=5 is 15.