Number series & sequences

Each answer names the pattern type (arithmetic / geometric / growing-gap / Fibonacci / interleaved) before the arithmetic.

Timed drill

(c) 25. Arithmetic, gap +5.
(c) 64. Geometric, ratio $\times 2$ . $32 \times 2 = 64$ .
(b) 26. Arithmetic, gap -6. $32 - 6 = 26$ .
(c) 21. Growing gaps +2, +3, +4, +5, next +6. $15 + 6 = 21$ . (Triangular numbers.)
(b) 1. Geometric, ratio $\div 3$ . $3 \div 3 = 1$ .
(b) 10. Arithmetic, gap +3. $7 + 3 = 10$ .
(c) 36. Squares: $1^2, 2^2, 3^2, 4^2, 5^2, 6^2$ . Next is $36$ .
(c) 12.5. Geometric, ratio $\div 2$ . $25 \div 2 = 12.5$ .
(c) 37. Gaps +3, +5, +7, +9, next +11. $26 + 11 = 37$ .
(c) 48. Geometric, ratio $\times 2$ . $24 \times 2 = 48$ .
(c) 13. Fibonacci: add the previous two. $5 + 8 = 13$ .
(b) 8. Arithmetic, gap -3. $11 - 3 = 8$ .
(c) 54. Geometric, ratio $\times 3$ . $18 \times 3 = 54$ and $54 \times 3 = 162$ checks.
(b) 40. Second value = first $\times 10$ . $4 \times 10 = 40$ .
(c) 125. Cubes: $1, 2^3, 3^3, 4^3, 5^3$ . Next is $125$ .
(c) 70. Interleaved. Odd positions 1, 2, 3, 4 (+1). Even positions 100, 90, 80, ? (-10). Next even term $= 70$ .
(c) 63. Rule $\times 2 + 1$ : $3 \to 7 \to 15 \to 31 \to 63$ .
(c) 80. Geometric, ratio $\times 2$ . $40 \times 2 = 80$ and $80 \times 2 = 160$ .
(c) 66. Gaps -10, -9, -8, next -7. $73 - 7 = 66$ .
(b) 34. Fibonacci-style: $13 + 21 = 34$ .

(d) 720. Ratios grow: $\times 2, \times 3, \times 4, \times 5$ , next $\times 6$ . $120 \times 6 = 720$ . (Factorials: $1!, 2!, 3!, 4!, 5!, 6!$ .)
(c) 33. Rule $\times 2 - 1$ : $2, 3, 5, 9, 17, 33$ .
(c) 16. Squares. Missing is $4^2 = 16$ .
(b) 63. Rule $\times 2 + 1$ : $31 \times 2 + 1 = 63$ .
(b) 42. Gaps +4, +6, +8, +10, next +12. $30 + 12 = 42$ . (Equivalently $n(n+1)$ .)
(b) 4. Geometric halving. $8 \div 2 = 4$ , and $4 \div 2 = 2$ checks.
(b) 42. First differences 1, 2, 4, 7, 11; second differences 1, 2, 3, 4, next 5. So next first difference $= 11 + 5 = 16$ , and $26 + 16 = 42$ .
(b) 10. Interleaved. Odd positions 1, 2, 3, 4, 5 (+1). Even positions 2, 4, 6, 8, ? (+2). Next even term $= 10$ .