Question

mrs. newman’s sorting. 4 cards 6,11,16,21,26,.... 2,5,8,11,14,.... 30,26,22,18,.... 40,43,46,49,.... 4 cards 100,50,25,12.5,.... 99,33,11, \frac{11}{3},.... 1,5,25,125,.... 8,16,32,64,.... mrs. newman grouped the sequences as shown in the picture. why do you think she did this?

Explanation:

Step1: Analyze first group of sequences

First group (top 4 cards):

• Sequence 1: \(6, 11, 16, 21, 26, \dots\) Check difference: \(11 - 6 = 5\), \(16 - 11 = 5\), \(21 - 16 = 5\), \(26 - 21 = 5\). Arithmetic sequence with common difference \(d = 5\).
• Sequence 2: \(2, 5, 8, 11, 14, \dots\) Difference: \(5 - 2 = 3\), \(8 - 5 = 3\), \(11 - 8 = 3\), \(14 - 11 = 3\). Arithmetic sequence with \(d = 3\).
• Sequence 3: \(30, 26, 22, 18, \dots\) Difference: \(26 - 30 = -4\), \(22 - 26 = -4\), \(18 - 22 = -4\). Arithmetic sequence with \(d = -4\).
• Sequence 4: \(40, 43, 46, 49, \dots\) Difference: \(43 - 40 = 3\), \(46 - 43 = 3\), \(49 - 46 = 3\). Arithmetic sequence with \(d = 3\).

So first group has arithmetic sequences (common difference between terms).

Step2: Analyze second group of sequences

Second group (bottom 4 cards):

• Sequence 1: \(100, 50, 25, 12.5, \dots\) Ratio: \(50\div100 = 0.5\), \(25\div50 = 0.5\), \(12.5\div25 = 0.5\). Geometric sequence with common ratio \(r = 0.5\).
• Sequence 2: \(99, 33, 11, \frac{11}{3}, \dots\) Ratio: \(33\div99=\frac{1}{3}\), \(11\div33=\frac{1}{3}\), \(\frac{11}{3}\div11=\frac{1}{3}\). Geometric sequence with \(r=\frac{1}{3}\).
• Sequence 3: \(1, 5, 25, 125, \dots\) Ratio: \(5\div1 = 5\), \(25\div5 = 5\), \(125\div25 = 5\). Geometric sequence with \(r = 5\).
• Sequence 4: \(8, 16, 32, 64, \dots\) Ratio: \(16\div8 = 2\), \(32\div16 = 2\), \(64\div32 = 2\). Geometric sequence with \(r = 2\).

So second group has geometric sequences (common ratio between terms).

Step3: Conclusion on grouping

Mrs. Newman grouped sequences by their type: arithmetic (first group, constant difference) and geometric (second group, constant ratio). This helps in teaching or analyzing sequence patterns by their core characteristic (arithmetic vs geometric progression).

Answer:

Mrs. Newman grouped the sequences by their type: the first group (top 4 cards) contains arithmetic sequences (with a constant common difference between consecutive terms), and the second group (bottom 4 cards) contains geometric sequences (with a constant common ratio between consecutive terms). This grouping helps in identifying and analyzing the fundamental pattern (arithmetic vs geometric progression) of each sequence.