Find the next number in the series 18, 6, 11, 32, 111, 464, ?

Concept: In many number-series items, each new term follows from the current term by the rule term(n + 1) = n × (term(n) + 5), where the multiplier n increases by exactly one at every step. Once this rule holds across several consecutive transitions in a series, the same rule can be carried one more step forward to find the missing term.

1. n = 1: 1 × (6 + 5) = 11, matching the third listed term.

2. n = 2: 2 × (11 + 5) = 32, matching the fourth listed term.

3. n = 3: 3 × (32 + 5) = 111, matching the fifth listed term.

4. n = 4: 4 × (111 + 5) = 464, matching the sixth listed term.

5. n = 5: 5 × (464 + 5) = 5 × 469 = 2345, the required next term.

Cross-check: writing the same rule in its expanded multiply-then-add form, term(n + 1) = n × term(n) + 5n, gives 5 × 464 + 5 × 5 = 2320 + 25 = 2345 for this final step — the two equivalent forms of the rule agree.

Note: the leading value 18 does not take part in this recurrence; the rule is confirmed across four independent, consecutive transitions among the remaining terms, which is sufficient to extend it reliably for the missing fifth transition.

Result: the next number in the series is 2345.