How does:
(1 + 2 + ... + N) / N = (N + 1) / 2
or
(1 + 2 + ... + N + N) / N = (N + 3) / 2
My textbook says this is elementary math but I have forgotten the method for finding the answer.
+2
A:
Mathematical induction. http://en.wikipedia.org/wiki/Mathematical_induction#Example
The second claim you stated follows from the first by just adding N / N = 1 = 2 / 2.
MrMage
2010-01-19 07:03:33
+8
A:
The example you gave is called an arithmetic sequence, not a geometric sequence.
A simple way to convince yourself that the result is correct is to write the same sequence backwards and add it to itself:
1 + 2 + ... + N-1 + N
N + N-1 + ... + 2 + 1
---------------------------
N+1 + N+1 + ... + N+1 + N+1
= (N+1)*N
This is twice the result, so divide by 2.
Mark Byers
2010-01-19 07:04:36
Ah this is the trick I was looking for. Thank you.
Brandon
2010-01-19 07:06:18
good answer! :)
Darmen
2010-01-19 07:06:26
But this won't work for all sequences so use with caution.
Vinko Vrsalovic
2010-01-19 07:09:57
It works only with finite arithmetic sequences.
Mark Byers
2010-01-19 07:11:33
That method is often attributed to a very young Gauss: http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#Early_years_.281777.E2.80.931798.29
Seth
2010-01-19 07:13:41
A:
Sum of n natural numbers is denoted by n(n+1)/2.
So the given the first problem you have correctly mentioned the output will (n+1)/2.
for the second problem.
the solution is (n(n+1)/2n)+n/n = (n+1)/2 +1 = (n+3)/2. You would observer actual series is sum of n natural numbers plus n. So thats how i split the terms.
Kavitesh Singh
2010-01-19 07:05:51