How do you get N(N+1) from N+1 + N + 1 + ... + N + 1 + N + 1?

Question

How does:

    1 +   2 + ... + N-1 +   N
 +  N + N-1 + ... +   2 +   1
  ---------------------------
 N+1 + N+1 + ... + N+1 + N+1

equal N(N + 1)? Shouldn't it be 4N + 4 or 4(N + 1)?

Answer 1

If N is 4, sure. Otherwise you need to fill in the rest of the elided values that the ellipses represent.

Answer 2

It is N(N + 1).

Because you have N number of (N+1) terms.

Answer 3

i assume your notation means row 1 + row 2 = row 3?

in this case, look at the columns. Each column of the first 2 rows adds up to n+1. there are n columns. thus row 1 + row 2 = n*(n+1)

Answer 4

Read the part about the early years of Carl Friederich Gauss here. He solved almost the same problem when he was in primary school.