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Does the general formula obtained by recursive formula need to be tested by mathematical induction?
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.