Why must add 1

It’s the formula. Without the 1, the answer is different and incorrect.

Note that P (r/100)^n is an incorrect way of obtaining the answer, which is not the same as P (1 + r/100)^n minus P.

You place 2500 into the bank paying 15%.

At the end of the first year, new final sum is 115% x 2500 = 2875.

In the second year, the 15% acts on the new sum of 2875 (that’s how compound interest works).

At the end of the second year, new sum is 115% x 2875 = 3306.25.

After the second year, the 15% acts on the new sum of 3306.25.

At the end of the third year, new sum is 115% x 3306.25 = 3802.1875.

Total interest accumulated along the way is final sum - initial sum = 3802.1875 - 2500 = 1302.1875, which rounds off (to the nearest dollar) to 1302.

Incorrect is

2500 x 0.15^3 = 8.4375

This is worse than having 15% interest for one year!

Does this make sense?