Which of the following numbers will replace the question marks (?) in the given series? 4, ?, ?, 49, 121, 169, 289

Correct — A. Every printed term is a perfect square, so take square roots before anything else.

4 = 2², 49 = 7², 121 = 11², 169 = 13², 289 = 17²
Roots so far: 2, _, _, 7, 11, 13, 17

Rule: the squares of consecutive prime numbers. The primes are 2, 3, 5, 7, 11, 13, 17.

The two missing roots are therefore 3 and 5.
3² = 9 and 5² = 25 — option (a).

Why the others are wrong

• (b)
  3, 9 — Option (b) inserts 3 and 9, but 3 is not a perfect square and every other term in the series is one — 4, 49, 121, 169 and 289.
• (c)
  9, 16 — Option (c) has 9 right and then takes 16, which is 4². 4 is not prime, so the root sequence would read 2, 3, 4, 7 and skip 5 altogether.
• (d)
  16, 25 — Option (d) gives roots of 4 and 5. 4 is not prime, and this reading also drops 3 out of the run of primes between 2 and 7.

When every visible term of a series is a perfect square, the series is really about the roots. Take them and the question usually collapses to something simple.

Here the roots are 2, then two gaps, then 7, 11, 13 and 17. Those are all primes, and 11, 13, 17 are consecutive primes — so the family is squares of consecutive primes and the gaps are 3² and 5².

Squares grow fast enough to look irregular, which is the trap. The jump from 4 to 49 is 45, but in root terms it is only 2 to 7.

Key facts

• The completed series is 4, 9, 25, 49, 121, 169, 289.
• Their square roots are 2, 3, 5, 7, 11, 13 and 17, which are the first seven prime numbers.
• 1 is not a prime number, which is why the series starts at 2² and not at 1².

Common traps

• Reading the roots as consecutive integers and inserting 16 for 4²
• Treating 1 as a prime and shifting the whole root sequence by one place
• Filling a blank with 3, which breaks the all-squares property the rest of the series has

No directly related past PYQ was found.