Question

If a sequence contains only whole numbers, which of these sequences could include the term 27, if a1 does not equal 27?

an = 2an – 1

an = 2an – 1 + 1

an = 3an – 1

an = an – 1 + 6

an = 3an – 1 + 1

Answer 1

Only the recurrences
- a_n = 2 a_{n-1} + 1,
- a_n = 3 a_{n-1},
- a_n = a_{n-1} + 6

can produce 27 for some whole-number a1 ≠ 27.

Reason:
- a_n = 2 a_{n-1}: any term for n≥2 is even, so cannot be 27 (odd). a1 = 27 is disallowed.
- a_n = 2 a_{n-1} + 1: choose a1 = 13 → a2 = 2·13 + 1 = 27.
- a_n = 3 a_{n-1}: choose a1 = 9 → a2 = 3·9 = 27.
- a_n = a_{n-1} + 6: choose a1 = 21 → a2 = 21 + 6 = 27.
- a_n = 3 a_{n-1} + 1: (27 − 1)/3 = 26/3 not integer, so 27 cannot be any term except a1 (disallowed).

Thus the 2nd, 3rd and 4th recurrences work.