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 (j1) = 1 + 2 + 3 + 4 + 5 + ... + n1
and this is the arithmetic progression n * ( N1) / 2
Answer 3
and formula 1) and formula 2) can be proved with the help of mathematical induction

We will verify loyalty at
n = 2
:Sum = J  1 = 2  1 = 1 = 2 (21) / 2 = 1
– True 
Let the formula are true for n, check it with
k = n + 1
, we getn (n  1) / 2 + n = (k1) (k2) / 2 + (k  1) = (k ^ 2 3k + 2 = 2k  2) / 2 = k (k1) / 2  right
Formula proved