Thinking Mathematically: Inductive Reasoning, Deductive Reasoning, and Patterns

Inductive and Deductive Reasoning

. Inductive Reasoning: This is a line of argument that begins with specific examples as its premise and draws a general conclusion based on those examples. It moves from small, specific observations to a broad generalization.

.Deductive Reasoning: This line of reasoning uses statements and definitions commonly accepted as facts to make a case for a specific conclusion. It moves from the "big" (general facts) to the "small" (specific instance).

Examples:

a.) In Bolger, Colorado, it snowed $16.2$ inches during January $2019$ and $16.3$ inches during January 2020, therefore boulder will receive at least 16 inches of snow every January. ( Inductive Reasoning.)

b.) All even numbers are divisible by $2$. The number $34,680,304$ is even, therefore , it is divisible by $2$.( Deductive Reasoning.)

c.) To get a bachelor's degree at the University of Florida, a student must have $120$ credit hours. Shay is about to graduate from the University of Florida with a bachelor's degree in engineering, therefore Shay has at least $120$ credit hours on her transcript. ( Deductive Reasoning.)

d.) Rebecca Plays basketball and made $40$ out of every 70 free throws in her last 50 games. so, she is expected to make $8$ out of $14$ free throws in tonight's game. ( Deductive Reasoning.)

e.) If you live in New York, you are a resident of the United States. ( Deductive Reasoning.)

TYPES OF PATTERN/SEQUENCES.

. Arithmetic Sequence: A sequence whose difference between two consecutive terms is always the same. This constant value is known as the common difference, ( Adding the same number repeatedly.)

. Geometric Sequence: A sequence in which the quotient between two consecutive terms is always the same. This constant value is known as the common ratio. ( Dividing the same number repeatedly.)

Example 2:

a.) 4,6,8,12, 14 , 16 , 18 . [ $6 - 4 = 2$, $8 - 6 = 2$, $10 - 8 = 2$, $12 - 10 = 2$.]

Common Difference :$2$. ( Arithmetic Sequence).

b.) o, -3, -6, -9, -12 -15 , -18 , -24 [ $-3 - 0 = -3$, $-6 - (-3) = -3$, $-12 - (-9) = -3$.]

• Common Difference : $-3$. This indicates the sequence is going down by $3$ for each step. (Arithmetic sequence.)

• c.) 5, 10, 20, 40, 80, 160, 320 [ $10 / 5 = 2$ , $20 / 10 = 2$, $40 / 20 = 2$.]

  • Common Ratio = $2$. Each term is multiplied by $2$ to get the next term. (Geometric sequence.)

• d.) 1, 4, 9, 16, 25, 36, 49, 64. [ The differences are not constant ($3, 5, 7, 9$), and the quotients are not constant ($4 / 1 = 4$, $9 / 4 = 2.25$).] ( This sequence consists of perfect squares: $1^2, 2^2, 3^2, 4^2, 5^2$. Alternatively, the differences between terms increase by $2$ each time ($3, 5, 7, 9$).)

• e.) 3, 9, 27, 81, 243, 729, 2187, 6517. [ $9 / 3 = 3$, $27 / 9 = 3$, $81 / 27 = 3$, $243 / 81 = 3$.]

• Common Ratio : $3$. (: Geometric sequence.)

  • Calculations for Next Terms:

    • $243 \times 3 = 729$

    • $729 \times 3 = 2,187$

    • $2,187 \times 3 = 6,561$

  • Try It Yourself;

  • * -3, 6, -12, 24, -48, 96, -192.

  • Multiply by (-2) to get the next terms. Common Ratio= -2 ( Type of sequence ; Geometric Sequence)

Counterexamples

• : A counterexample is a specific example that proves a statement is false.

• Example:

• * All mammals live on land, While cats, dogs, humans, horses, cows, and sheep are land-dwelling mammals, there are mammals that live in water. Counterexamples: Dolphins, whales, and seals are mammals that do not live on land. Their existence makes the statement FALSE.

a.) The difference between two odd numbers is an odd number. FALSE

, 11...$</p><ul><li><p><strong>Counterexamples</strong>:</p><ul><li><p>$21 - 1 = 20$</p></li><li><p>$35 - 5 = 30$</p></li></ul></li></ul><p>b.) The product of two even numbers is always even. <strong><u>TRUE</u></strong></p><ul><li><p>$2 \times 10 = 20$</p></li><li><p>$10 \times 12 = 120$</p></li><li><p>$6 \times 4 = 24$</p></li><li><p>$8 \times 8 = 64$</p></li><li><p>$100 \times 2 = 200$</p></li></ul><p>c.) If a geometric figure has four sides, then the figure is a square.<strong><u> FALSE</u></strong></p><ul><li><p><strong>Definitions</strong>: A square must have four sides of exactly the same size and four$90^{\circ}$(right) angles.</p></li><li><p><strong>Counterexample</strong>: A rectangle or any four-sided polygon where the sides are not equal or the angles are not right angles. Because these figures have four sides but are not squares, the statement is false.</p></li></ul><h4>Discussion and Logic Exercises</h4><ul><li><p><strong>TRY IT YOURSELF:</strong></p></li><li><p>d.) All the students in this class were born in Miami, Florida. <strong>FALSE .COUNTEREXAMPLE: </strong>If there are$28$students in a class, it is highly probable at least one was born elsewhere, Prince was born in Nigeria a totally different Country.</p></li><li><p>e.) When multiplying a number by$2$$, the result is always an even number. TRUE