Arithmetic Sequences Challenge Problem
Whole Numbers & Operations Unit – Problem #2
Consider the following two sequences:
5,14,23,32,41,50…
20, 27,34, 41,48,55…
and
Note that 41 is common to both sequences. Find the next ten numbers which are common to
both sequences. Look for and explain at least four patterns in your new set and the two original
sets.
Sequences 1 & 2 are both arithmetic sequences, in that each number increases by a constant
interval.
5
14
+9
Sequence 1
20
Sequence 2
23
+9
27
+7
32
+9
34
+7
41
+9
41
+7
50
+9
48
+7
55
+7
Knowing this, let’s determine the equations for each of these sequences. We will denote
sequence 1 with the letter “a” and sequence 2 with the letter “b.”
Sequence 1: d = 9, a1 = 5
 an = 9(n-1) + 5
= 9n – 9 + 5
= 9n – 4
Sequence 2: d = 7, b1 = 20
 bn = 7(n-1) + 20
= 7n – 7 + 20
= 7n + 13
When n = 5 for an and n = 4 for bn, both of our equations will produce the number 41.
To find other numbers that are common to both sequences, we can program excel to produce
tables of our equations and then identify common numbers.
n
1
2
3
4
Sequence 1
5
14
23
32
Sequence 2
20
27
34
41
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
41
50
59
68
77
86
95
104
113
122
131
140
149
158
167
176
185
194
203
212
221
230
239
248
257
266
275
284
293
302
311
320
329
338
347
356
365
374
383
392
401
410
419
48
55
62
69
76
83
90
97
104
111
118
125
132
139
146
153
160
167
174
181
188
195
202
209
216
223
230
237
244
251
258
265
272
279
286
293
300
307
314
321
328
335
342
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
428
437
446
455
464
473
482
491
500
509
518
527
536
545
554
563
572
581
590
599
608
617
626
635
644
653
662
671
680
689
698
707
716
725
734
743
752
761
770
779
788
797
806
349
356
363
370
377
384
391
398
405
412
419
426
433
440
447
454
461
468
475
482
489
496
503
510
517
524
531
538
545
552
559
566
573
580
587
594
601
608
615
622
629
636
643
Looking at these common numbers as their own series, they too make an arithmetic sequence.
The common difference for this new sequence is 63, which is the common differences from
sequence 1 & 2 multiplied together!
 9 x 7 = 63
41
Sequence 3
104
+63
167
+63
230
+63
293
+63
Knowing this, we can find an equation for the sequence.
Sequence 3: d = 63, c1 = 41

c1 = 63(n-1) + 41
= 63n – 63 + 41
= 63n – 22
n
1
2
3
4
5
6
7
8
9
10
Sequence 3
41
104
167
230
293
356
419
482
545
608
356
+63