121, 143, 165, 187, ?

A number-series question is solved by identifying the single rule connecting every consecutive pair of given terms — commonly a constant additive step (an arithmetic progression) or a constant multiplicative step. Once that rule holds across all the given terms, applying it once more to the last term produces the missing term.

1. Find the difference between each consecutive pair of given terms: 143 − 121 = 22, 165 − 143 = 22, and 187 − 165 = 22.

2. All three differences are equal to 22, so the series is an arithmetic progression with a constant common difference of 22.

3. Apply the same common difference to the last given term: 187 + 22 = 209.

Cross-check using a second method — write each term as 11 times an odd number:

• 121 = 11 × 11

• 143 = 11 × 13

• 165 = 11 × 15

• 187 = 11 × 17

The multipliers 11, 13, 15, 17 increase by 2 each time — consistent with the common difference of 22 (= 11 × 2). Note these are consecutive odd numbers, not prime numbers, since 15 is not prime. Continuing the pattern, the next multiplier is 19, giving 11 × 19 = 209 — the same result obtained from the first method.

So the missing term in the series is 209.