Question

sort the sequences into two groups - there is no right or wrong answer as long as you have a reason for how you sorted them!
40, 43, 46, 49, ...
6, 11, 16, 21, 26, ...
30, 26, 22, 18, ...
2, 5, 8, 11, 14, ...
100, 50, 25, 12.5, ...
8, 16, 32, 64, ...
1, 5, 25, 125, ...
99, 33, 11, \\(\frac{11}{3}\\), ...

Explanation:

Step1: Identify Arithmetic Sequences

Check the common difference between consecutive terms.

• \(40, 43, 46, 49, \dots\): \(43 - 40 = 3\), \(46 - 43 = 3\), \(49 - 46 = 3\) (common difference \(d = 3\))
• \(6, 11, 16, 21, 26, \dots\): \(11 - 6 = 5\), \(16 - 11 = 5\), \(21 - 16 = 5\), \(26 - 21 = 5\) (common difference \(d = 5\))
• \(30, 26, 22, 18, \dots\): \(26 - 30 = -4\), \(22 - 26 = -4\), \(18 - 22 = -4\) (common difference \(d = -4\))
• \(2, 5, 8, 11, 14, \dots\): \(5 - 2 = 3\), \(8 - 5 = 3\), \(11 - 8 = 3\), \(14 - 11 = 3\) (common difference \(d = 3\))
• \(99, 33, 11, \frac{11}{3}, \dots\): \(33\div99=\frac{1}{3}\), \(11\div33=\frac{1}{3}\), \(\frac{11}{3}\div11=\frac{1}{3}\) (common ratio \(r=\frac{1}{3}\), but wait, let's re - check. Wait, \(33 - 99=-66\), no, \(33=\frac{1}{3}\times99\), \(11 = \frac{1}{3}\times33\), so it's a geometric sequence. Wait, maybe I made a mistake. Let's re - evaluate each sequence:

1. \(40, 43, 46, 49,\dots\): Arithmetic, \(d = 3\)
2. \(6, 11, 16, 21, 26,\dots\): Arithmetic, \(d = 5\)
3. \(30, 26, 22, 18,\dots\): Arithmetic, \(d=- 4\)
4. \(2, 5, 8, 11, 14,\dots\): Arithmetic, \(d = 3\)
5. \(8, 16, 32, 64,\dots\): Geometric, \(r = 2\) (since \(16\div8 = 2\), \(32\div16=2\), \(64\div32 = 2\))
6. \(100, 50, 25, 12.5,\dots\): Geometric, \(r=\frac{1}{2}\) (since \(50\div100=\frac{1}{2}\), \(25\div50=\frac{1}{2}\), \(12.5\div25=\frac{1}{2}\))
7. \(1, 5, 25, 125,\dots\): Geometric, \(r = 5\) (since \(5\div1 = 5\), \(25\div5 = 5\), \(125\div25=5\))
8. \(99, 33, 11,\frac{11}{3},\dots\): Geometric, \(r=\frac{1}{3}\) (since \(33\div99=\frac{1}{3}\), \(11\div33=\frac{1}{3}\), \(\frac{11}{3}\div11=\frac{1}{3}\))

Let's sort them into arithmetic and geometric sequences.

Arithmetic sequences (where the difference between consecutive terms is constant):

• \(40, 43, 46, 49,\dots\)
• \(6, 11, 16, 21, 26,\dots\)
• \(30, 26, 22, 18,\dots\)
• \(2, 5, 8, 11, 14,\dots\)

Geometric sequences (where the ratio between consecutive terms is constant):

• \(8, 16, 32, 64,\dots\)
• \(100, 50, 25, 12.5,\dots\)
• \(1, 5, 25, 125,\dots\)
• \(99, 33, 11,\frac{11}{3},\dots\)

Step2: Define the sorting rule

We can sort the sequences into two groups: Arithmetic Sequences and Geometric Sequences.

Group 1 (Arithmetic Sequences):

• \(40, 43, 46, 49,\dots\)
• \(6, 11, 16, 21, 26,\dots\)
• \(30, 26, 22, 18,\dots\)
• \(2, 5, 8, 11, 14,\dots\)

Group 2 (Geometric Sequences):

• \(8, 16, 32, 64,\dots\)
• \(100, 50, 25, 12.5,\dots\)
• \(1, 5, 25, 125,\dots\)
• \(99, 33, 11,\frac{11}{3},\dots\)

Answer:

Group 1 (Arithmetic Sequences): \(40, 43, 46, 49,\dots\); \(6, 11, 16, 21, 26,\dots\); \(30, 26, 22, 18,\dots\); \(2, 5, 8, 11, 14,\dots\)

Group 2 (Geometric Sequences): \(8, 16, 32, 64,\dots\); \(100, 50, 25, 12.5,\dots\); \(1, 5, 25, 125,\dots\); \(99, 33, 11,\frac{11}{3},\dots\)

(The sorting is based on the type of sequence, arithmetic sequences have a constant difference between consecutive terms and geometric sequences have a constant ratio between consecutive terms)