Find the missing term: 7, 22, 89,________, 2677, 18740

Correct — A.
Rule: multiply by 3, 4, 5, 6, 7 in turn, adding 1 each time.

7 × 3 + 1 = 22
22 × 4 + 1 = 89
89 × 5 + 1 = 446
446 × 6 + 1 = 2677 ✓

2677 × 7 + 1 = 18740 ✓, so the missing term is 446 → option (a).

Why the others are wrong

• (b)
  436 — 436 is 10 short of 89 × 5 + 1 = 446. It also fails forwards: 436 × 6 + 1 = 2617, not 2677.
• (c)
  448 — 448 is 2 too many: 89 × 5 + 1 = 446. And 448 × 6 + 1 = 2689, which misses 2677.
• (d)
  457 — 457 fits neither side: 89 × 5 + 1 = 446, and 457 × 6 + 1 = 2743, not 2677.

When the terms grow several times over at each step, 7 to 22 to 89, look at ratios, not differences. 22 ÷ 7 is just over 3 and 89 ÷ 22 just over 4: a multiplier rising by one, plus a small constant.

Test the rule on a known pair before trusting it. Here 2677 × 7 + 1 = 18740 confirms it on the terms after the gap.

Key facts

• Rule of this series: next term = term × n + 1, with n = 3, 4, 5, 6, 7.
• A missing middle term can be checked from both sides: 89 → 446 and 446 → 2677.
• 2677 × 7 = 18739, and 18739 + 1 = 18740.

Common traps

• Working with differences (15, 67), which grow too fast to reveal the rule
• Checking 446 against 89 only and not against 2677 after the gap

No directly related past PYQ was found.