What will be the value of P in the following number series? 0, 3, 26, 255, P,…

Concept

In a number series the generating rule may tie each term to its own position n rather than to the term before it. One standard family is the power in which the base and the exponent move together with the position: the n-th term is nn − 1.

The way to spot this family is to add 1 to every given term and ask whether the result is a perfect power whose base equals that term’s position number.

Application

1. Position 1: 11 − 1 = 1 − 1 = 0, which is the first term.

2. Position 2: 22 − 1 = 4 − 1 = 3, which is the second term.

3. Position 3: 33 − 1 = 27 − 1 = 26, which is the third term.

4. Position 4: 44 − 1 = 256 − 1 = 255, which is the fourth term.

5. Position 5: P = 55 − 1 = 3125 − 1 = 3124.

Cross-check

• Add 1 to each listed term: 1, 4, 27, 256. These are exactly 11, 22, 33, 44, so the data itself fixes the rule nn − 1 with no free constant left to choose.

• Successive differences run 3, 23, 229 and then 3124 − 255 = 2869. Each difference is exactly (n+1)n+1 − nn, because the two −1 terms cancel when one term is subtracted from the next, so the growth pattern lands on 3124 as well.

• Values one less than a perfect power are common, so that alone is not enough: 624 = 54 − 1, 511 = 83 − 1 and 1023 = 322 − 1 all share that shape, but only 3124 = 55 − 1 has base and exponent both equal to the position 5.

Hence P = 3124.