# Thread: Difficult number sequences

1. ## Difficult number sequences

Anyone have any ideas or suggestions on how to finish these number sequences?

1) 01, 10, 0-1, ?
2) -1-1, 1-1, 11, ?
3) 236, 689, 478, ?
4) 4578, 1245, 2356, ?
5) 0, 3, 12, 30, 60, ?
6) 0.5, 2.5, 13, 32, ?
7) 1, 5, 19, 65, ?
8) 4651, 369, 37, 5, ?
9) 3, 2, 0.6, 0.4, 0.2, ?
10) 24, 20, 17, 14, ?
11) 5.5, 10, 16, 25, 40, ?
12) 16, 24, 36, 54, 81, ?
13) 5, 3, 2, 2, ?
14) 3, 2, 2, 2, 2, 2, ?
15) 0, 6, 20, 42, ?
16) 0, 2, 28, 30, ?
17) 6, 2, 5, 5, 4, ?
18) 1, 2, 3, 4, 2, ?
19) A, 13, B, 5, C, 3, D, 23, E, ?
20) F, 3, G, 2, H, 19, I, 2, J, ?
21) 0, 1, 0.5, 0.5, 0, 0, 0.75, ?
22) 0, 0, 0, 0, 0, 0.5, 0, ?
23) 225, 37, 211, 23, ?
24) 235, 29, 227, 333, ?
25) 12, 111, 210, ?
26) 3, 24, 15, 126, ?
27) 3, 14, 159, ?
28) 2, 71, 828, ?

2. Originally Posted by dxdy
Anyone have any ideas or suggestions on how to finish these number sequences?

1) 01, 10, 0-1, ?
2) -1-1, 1-1, 11, ?
3) 236, 689, 478, ?
4) 4578, 1245, 2356, ?
5) 0, 3, 12, 30, 60, ?
6) 0.5, 2.5, 13, 32, ?
7) 1, 5, 19, 65, ?
8) 4651, 369, 37, 5, ?
9) 3, 2, 0.6, 0.4, 0.2, ?
10) 24, 20, 17, 14, ?
11) 5.5, 10, 16, 25, 40, ?
12) 16, 24, 36, 54, 81, ?
13) 5, 3, 2, 2, ?
14) 3, 2, 2, 2, 2, 2, ?
15) 0, 6, 20, 42, ?
16) 0, 2, 28, 30, ?
17) 6, 2, 5, 5, 4, ?
18) 1, 2, 3, 4, 2, ?
19) A, 13, B, 5, C, 3, D, 23, E, ?
20) F, 3, G, 2, H, 19, I, 2, J, ?
21) 0, 1, 0.5, 0.5, 0, 0, 0.75, ?
22) 0, 0, 0, 0, 0, 0.5, 0, ?
23) 225, 37, 211, 23, ?
24) 235, 29, 227, 333, ?
25) 12, 111, 210, ?
26) 3, 24, 15, 126, ?
27) 3, 14, 159, ?
28) 2, 71, 828, ?
HERE is #5.
You can use that website to do any of those.

3. Hello, dxdy!

Most of them have too few terms to discern a pattern . . .

$\displaystyle 5)\;0,\:3,\:12,\:30,\:60,\:?$

. . $\displaystyle \begin{array}{c|| c|c|c|c|c|c|} n & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline a_n & 0 & 3 & 12 & 30 & 60 & \qquad\\ \hline \\[-4mm] & \frac{0\cdot1\cdot2}{2} & \frac{1\cdot2\cdot3}{2} & \frac{2\cdot3\cdot4}{2} & \frac{3\cdot4\cdot5}{2} & \frac{4\cdot5\cdot6}{2} & \end{array}$

$\displaystyle \text{The general term is: }\:A_n \;=\;\frac{(n-1)n(n+1)}{2}$

$\displaystyle \text{Therefore: }\:a_6 \:=\:\frac{5\cdot6\cdot7}{2} \:=\:105$

$\displaystyle 12)\;16,\:24,\:36,\:54,\:81,\:?$

. . $\displaystyle \begin{array}{c|| c|c|c|c|c|c|} n & \;\;1\;\; & \;2\; & 3 & 4 & 5 & 6 \\ \hline a_n & 16 & 24 & 36 & 54 & 81 & \qquad \\ \hline \\[-4mm] & 16 & 16(\frac{3}{2}) & 16(\frac{3}{2})^2 & 16(\frac{3}{2})^3 & 16(\frac{3}{2})^4 & \end{array}$

$\displaystyle \text{A geometric sequence with first term }a = 16\text{ and common ratio }r = \tfrac{3}{2}$

$\displaystyle \text{The general term is: }\:a_n \;=\;16(\tfrac{3}{2})^{n-1}$

$\displaystyle \text{Therefore: }\:a_6 \;=\;16(\tfrac{3}{2})^5 \:=\:\frac{243}{2}$

$\displaystyle 15)\;0,\:6,\:20,\:42,\:?$

. . $\displaystyle \begin{array}{c|| c|c|c|c|c|} n & 1 & 2 & 3 & 4 & 5 \\ \hline a_n & 0 & 6 & 20 & 42 & \qquad \\ \hline \\[-4mm] & 0\cdot1 & 2\cdot3 & 4\cdot5 & 6\cdot7 & \end{array}$

$\displaystyle \text{The general term is: }\:a_n \;=\;(2n-2)(2n-1) \:=\:2(n-1)(2n-1)$

$\displaystyle \text{Therefore: }\:a_5 \;=\;2(4)(9) \:=\:72$

$\displaystyle 25)\;12,\:111,\:210,\:?$

They seem to be numerals written in base-3.

. . $\displaystyle \begin{array}{c|| c|c|c|c|c|c|} n & 1 & 2 & 3 & 4 \\ \hline a_n & 12_3 & 111_3 & 210_3 & \qquad \\ \hline & 5_{10} & 13_{10} & 21_{10} & \\ \end{array}$

$\displaystyle \text{The base-ten numbers form an arithmetic sequence}$
. . $\displaystyle \text{with first term }a = 5\text{ and common difference }d = 8.$

$\displaystyle \text{The general term is: }\:a_n \:=\:5 + (n-1)8$

$\displaystyle \text{Therefore: }\:a_4 \:=\:5 + 3(8) \:=\:29 \;=\;1002_3$

4. Hello, dxdy!

Some of them are downright silly . . . like #14 and #22.

While others are rather "trivial" . . . like #3 and #10.

$\displaystyle 3)\;\;236,\:689,\:478,\:?$

There is always a quadratic function for three points: (1, 236), (2, 689), (3, 478)

One such function: .$\displaystyle a_n \:=\:-332n^2 + 1449n - 881$

Therefore: .$\displaystyle a_4 \:=\:-332(16) + 1449(4) - 881 \:=\:-397$

$\displaystyle 10)\;\;24,\:20,\:17,\:14\:?$

There is always a cubic function that passes through four points:
. . (1, 24), (2, 20), (3,17), (4,14).

One such function: .$\displaystyle a_n \;=\;-\tfrac{1}{6}n^3 + \tfrac{3}{2}n^2 - \tfrac{22}{3}n + 30$

Therefore: .$\displaystyle a_5 \;=\;-\tfrac{1}{6}(125) + \tfrac{3}{2}(25) - \tfrac{22}{3}(5) + 30 \;=\;15$

5. Originally Posted by dxdy
Anyone have any ideas or suggestions on how to finish these number sequences?

1) 01, 10, 0-1, ?
2) -1-1, 1-1, 11, ?
3) 236, 689, 478, ?
4) 4578, 1245, 2356, ?
5) 0, 3, 12, 30, 60, ?
6) 0.5, 2.5, 13, 32, ?
7) 1, 5, 19, 65, ?
8) 4651, 369, 37, 5, ?
9) 3, 2, 0.6, 0.4, 0.2, ?
10) 24, 20, 17, 14, ?
11) 5.5, 10, 16, 25, 40, ?
12) 16, 24, 36, 54, 81, ?
13) 5, 3, 2, 2, ?
14) 3, 2, 2, 2, 2, 2, ?
15) 0, 6, 20, 42, ?
16) 0, 2, 28, 30, ?
17) 6, 2, 5, 5, 4, ?
18) 1, 2, 3, 4, 2, ?
19) A, 13, B, 5, C, 3, D, 23, E, ?
20) F, 3, G, 2, H, 19, I, 2, J, ?
21) 0, 1, 0.5, 0.5, 0, 0, 0.75, ?
22) 0, 0, 0, 0, 0, 0.5, 0, ?
23) 225, 37, 211, 23, ?
24) 235, 29, 227, 333, ?
25) 12, 111, 210, ?
26) 3, 24, 15, 126, ?
27) 3, 14, 159, ?
28) 2, 71, 828, ?
Too many questions. The replies given tell you what the problem is with all of these.

Thread closed.