If we guess the general formula from the laws of the previous items, we need to verify it by induction. If this is the case, for example:

It is known that A 1= 1 and an -2A(n- 1)= 1.

This gives an/2n-a (n-1)/2 (n-1) =1/2n, so the following equation is added:

an/2^n-a(n- 1)/2^(n- 1)= 1/2^n

a(n- 1)/2^(n- 1)-a(n-2)/2^(n-2)= 1/2^(n- 1)

.........................

A2/2^2-A 1/2= 1/2^2

Get an = 2 n- 1。

The results don't need to be tested by induction.