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Learning Math Home
Patterns, Functions, and Algebra
 
Session 6 Part A Part B Part C Part D Part E Homework
 
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Session 2 Materials:
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A B C D E
Homework

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Solutions for Session 2, Part A

See solutions for Problems: A1 | A2 | A3 | A4 | A5 | A6


Problem A1

Some possible answers: each output number is 4 more than the last; the output numbers that appear are all the even numbers that aren't multiples of 4 (starting with 6); the output number is 2 more than 4 times the input number. Also, adding one input to the following input yields half the first output.

<< back to Problem A1


 

Problem A2

You can't be sure, because the pattern is not completely specified, but it would be likely that the 100th number is 402. This follows the third rule listed above -- that the output number is 2 more than 4 times the input number.

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Problem A3

Again, you can't be completely sure, but it would be likely that the 25th number is 102, because 2 more than 4 times 25 is 102. Following the same pattern, 1004 would not appear in the output column, since 1004 is not 2 more than 4 times any whole number.

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Problem A4

Example: Pick 7, and then follow the algorithm. 7 >> 21 >> 19 >> 38 >> 44 >> 30 is the output. The numbers at the end are the same as the pattern described in the table. Here's why: Pick n instead, which stands for a variable number. Follow the algorithm. n >> 3n >> 3n - 2 >> 6n - 4 >> 6n + 2 >> 4n + 2 is the output, which is the rule described in Problem A1.

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Problem A5

Some comments:

a. 

This describes the way the table is built, but doesn't say what values the table begins with.

b. 

If the first input is n, the sum of the first input and the next is n + (n + 1) = 2n + 1. Doubling this gives 4n + 2, which is the formula for the table.

c. 

This is a good digit-based description of the table.

d. 

It's 4n + 2 again. An efficient, closed-form description.

e. 

Triple the input is 3n, then 2 more than the input is n + 2, so the sum is 4n + 2.

f. 

There's no way of being sure that 4n + 2 is the correct pattern. It is definitely not the only pattern that starts 6, 10, 14, 18, 22, 26, ... .

g. 

This is another example of a different continuation to the pattern. Because the given table ends with 26, this is a valid continuation.

<< back to Problem A5


 

Problem A6

The next number could be any number with a justified pattern. 4 is the clear choice, but so is 5, the next Fibonacci number, or 10, the next number you would get when counting in base 4, or 1, the next beat in a waltz. A formula could be found for any 4th number in that sequence.

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