What is the next term in the series 4, 9, 25, 49, 121, ...?
SHORT ANSWER
169. The terms are squares of consecutive prime numbers: 2^2, 3^2, 5^2, 7^2, 11^2, so the next is 13^2 = 169.
Explain it simply
When terms grow unevenly, test whether they are well-known numbers first: squares, cubes, primes, factorials. Here every term is a perfect square, but the roots 2, 3, 5, 7, 11 skip 4, 6, 8, 9 and 10. The roots are prime numbers, so the next root is the next prime, 13.
Example
Question: 4, 9, 25, 49, 121, ?
- Square roots: 2, 3, 5, 7, 11.
- These are consecutive primes. The next prime is 13.
- Next term = 13^2 = 169.
Differences (5, 16, 24, 72) show no pattern, which is a signal to look at the terms themselves rather than the gaps.
Similar series:
- 8, 27, 125, 343, ? gives 11^3 = 1,331 (cubes of primes).
- 1, 2, 6, 24, 120, ? gives 720 (factorials).
js
const primes = [2, 3, 5, 7, 11, 13];
console.log(primes.map(p => p * p)); // Output: [4,9,25,49,121,169]What to say in the interview
"All the terms are perfect squares, and their roots, 2, 3, 5, 7, 11, are consecutive primes. The next prime is 13, so the next term is 169."
Common mistakes
- Answering 144 (12^2) by assuming the roots just increase.
- Hunting for patterns in the differences when the terms themselves are recognisable.