Hello, you can tell me where the formula of 2 lines is displayed. In 1 it is clear this is the sum of the first NAT. numbers, but why in 2 formula appears minus in brackets? Thank you.

Answer 1, Authority 100%

Actually, that’s all – the text is needed only to drop 30 characters 🙂

however, here’s another explanation:

Answer 2, Authority 100%

everything is simple to paint and get

Summ _ j = 2 ^ n (j-1) = 1 + 2 + 3 + 4 + 5 + ... + n-1 and this is the arithmetic progression n * ( N-1) / 2

Answer 3

and formula 1) and formula 2) can be proved with the help of mathematical induction

1. We will verify loyalty at n = 2 : Sum = J - 1 = 2 - 1 = 1 = 2 (2-1) / 2 = 1 – True

2. Let the formula are true for n, check it with k = n + 1 , we get

   n (n - 1) / 2 + n = (k-1) (k-2) / 2 + (k - 1) = (k ^ 2 -3k + 2 = 2k - 2) / 2 = k (k-1) / 2 - right
   

Formula proved