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Junior middle school mathematics
This topic looks very complicated, in fact, let's find the law.

(2+ 1)(2? +1) = 3+2 of 2? +2 (power of 2)+1(power of 2).

Then multiply 2+~+2+1by the square of 2+6.

In other words, this is the law. If it's a fill-in-the-blank question, that's it.

The answer is the (1+2+4+6+~+2n) power of 2 and the (1+2+4+6+~ 2n-1) power of 2+~+.

If this is a big problem, prove it.

There are many ways to prove it, so just assume that the above idea is correct, which is the result of the first n terms of 2n.

Then the term of n+ 1 is multiplied by another (2 2 (n+1)+1).

The result of the nth term is (1+2+4+6 +~+2n) power+2 (1+2+4+6+~ 2 (n-1) power+~+2.

If your idea is right, then the law of recursion is also right.

There is a quadratic sum of (1+2+4+6+~+2 (n+1)) and a quadratic sum of (1+2+4+6+~+2n+1).

Calculate the power (1+2+4+6+~+2n-1) of (1~+2+0).

Multiplied by (2 2 (n+1)+1) is equal to the power of (1 +2+4+6+~+2 (n+ 1)).

The original formula is proved.