Complete the pattern: 1, 4, 13, 40, 121, ?

Correct — A.
Rule: multiply by 3, then add 1.

1 × 3 + 1 = 4, 4 × 3 + 1 = 13
13 × 3 + 1 = 40, 40 × 3 + 1 = 121
121 × 3 + 1 = 363 + 1 = 364 → option (a).

Why the others are wrong

• (b)
  360 — 360 is 4 short of 121 × 3 + 1. Its gap from 121 would be 239, not the next power of 3, which is 243.
• (c)
  362 — 362 = 121 × 3 − 1. It subtracts 1 at the last step, where every earlier step adds 1.
• (d)
  365 — 365 = 121 × 3 + 2. It adds 2 at the last step, but 40 × 3 + 1 = 121 shows the constant is 1.

A series that grows about threefold at each step suggests a × 3 + c rule. Multiply each term by 3 and see what is left over: here it is 1 every time.

The gaps give a second route: 4 − 1 = 3, 13 − 4 = 9, 40 − 13 = 27, 121 − 40 = 81. They are powers of 3, so the next gap is 243, and 121 + 243 = 364.

Two routes reaching one number is your check.

Key facts

• 1, 4, 13, 40, 121 follows × 3 + 1 at every step.
• The differences 3, 9, 27, 81 are powers of 3, so the next difference is 243.
• 121 × 3 + 1 = 364, and 121 + 243 = 364.

Common traps

• Changing the constant at the last step: 121 × 3 − 1 = 362 and 121 × 3 + 2 = 365 are both on offer
• Checking the rule on the last two terms only instead of on every step from 1 → 4

No directly related past PYQ was found.