Pick any number, multiply the number by 4, add 12 to the product, divide the sum by 4, and...

Pick any number, multiply the number by 4, add 12 to the product, divide the sum by 4, and subtract 3 from the quotient. See Example.

a) What is the relationship between the number you started with and the final number?

b) Arbitrarily select some different numbers and repeat the process, recording the original number and the result.

c) Can you make a conjecture about the relationship between the original number and the final number?

d) Try to prove, using deductive reasoning, the conjecture you made in part (c). See Example.

Example

Pick a Number, Any Number

Pick any number, multiply the number by 4, add 2 to the product, divide the sum by 2, and subtract 1 from the quotient. Repeat this procedure for several different numbers and then make a conjecture about the relationship between the original number and the final number.

Solution

Let’s go through this one together.

Note that we started with the number 5 and finished with the number 10. If you start with the number 2, you will end with the number 4. Starting with 3 would result in a final number of 6, 4 would result in 8, and so on. On the basis of these few examples, we may conjecture that when you follow the given procedure, the number you end with will always be twice the original number.

Example

Pick a Number, n

Prove, using deductive reasoning, that the procedure in Example will always result in twice the original number selected.

Solutio To use deductive reasoning, we begin with the general case rather than specific examples. Specific cases were used. Let’s select the letter nto represent any number.

Note that for any number nselected, the result is 2n, or twice the original number selected.