Discrete Mathematics

Prove that 2−2·7+2·7^2 −···+2(−7)^n =(1− (−7)^(n+1))/4 whenever n is a nonnegative integer

Mathematical Induction /

To make this proof, we will use the principle of mathematical induction. PRINCIPLE OF MATHEMATICAL INDUCTION: “To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: BASIS STEP: We verify that P (1) is true. INDUCTIVE STEP: We show that the conditional statement […]

Prove that 2−2·7+2·7^2 −···+2(−7)^n =(1− (−7)^(n+1))/4 whenever n is a nonnegative integer Read More »

Let P(n) be the statement that 1^3 +2^3 +···+n^3 = (n(n + 1)/2)^2 for the positive integer n

Mathematical Induction /

PRINCIPLE OF MATHEMATICAL INDUCTION: “To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: BASIS STEP: We verify that P (1) is true. INDUCTIVE STEP: We show that the conditional statement P (k) → P (k + 1) is true for all positive

Let P(n) be the statement that 1^3 +2^3 +···+n^3 = (n(n + 1)/2)^2 for the positive integer n Read More »

a) Find a formula for 1/2 + 1/4 + 1/8 + … + 1/2^n by examining the values of this expression for small values of n. b) Prove the formula you conjectured in part (a).

Mathematical Induction /

We will use the mathematical induction to answer b). So, let’s start with the definition of the principle of mathematical induction. PRINCIPLE OF MATHEMATICAL INDUCTION: “To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: BASIS STEP: We verify that P (1) is

a) Find a formula for 1/2 + 1/4 + 1/8 + … + 1/2^n by examining the values of this expression for small values of n. b) Prove the formula you conjectured in part (a). Read More »

Prove that 3+3 · 5+3 · 5^2+···+3 · 5^n=3(5^(n+1)−1)/4 whenever n is a nonnegative integer

Mathematical Induction /

PRINCIPLE OF MATHEMATICAL INDUCTION: “To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: BASIS STEP: We verify that P (1) is true. INDUCTIVE STEP: We show that the conditional statement P (k) → P (k + 1) is true for all positive

Prove that 3+3 · 5+3 · 5^2+···+3 · 5^n=3(5^(n+1)−1)/4 whenever n is a nonnegative integer Read More »

Let P(n) be the statement that 1^2 +2^2 +···+n^2 = n(n + 1)(2n + 1)/6 for the positive integer n.

Mathematical Induction /

PRINCIPLE OF MATHEMATICAL INDUCTION: “To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: BASIS STEP: We verify that P (1) is true. INDUCTIVE STEP: We show that the conditional statement P (k) → P (k + 1) is true for all positive

Let P(n) be the statement that 1^2 +2^2 +···+n^2 = n(n + 1)(2n + 1)/6 for the positive integer n. Read More »

There are infinitely many stations on a train route. Suppose that the train stops at the first station and suppose that if the train stops at a station, then it stops at the next station. Show that the train stops at all stations

Mathematical Induction /

Let’s revisit the principle of mathematical induction. PRINCIPLE OF MATHEMATICAL INDUCTION: “To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: BASIS STEP: We verify that P (1) is true. INDUCTIVE STEP: We show that the conditional statement P (k) → P (k

There are infinitely many stations on a train route. Suppose that the train stops at the first station and suppose that if the train stops at a station, then it stops at the next station. Show that the train stops at all stations Read More »

How many different three-letter initials with none of the letters repeated can people have?

Counting /

Relevant definitions for this exercise: THE PRODUCT RULE: “Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do the second task, then there are

How many different three-letter initials with none of the letters repeated can people have? Read More »

There are four major auto routes from Boston to Detroit and six from Detroit to Los Angeles. How many major auto routes are there from Boston to Los Angeles via Detroit?

Counting /

Relevant definitions for this exercise: THE PRODUCT RULE: “Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do the second task, then there are

There are four major auto routes from Boston to Detroit and six from Detroit to Los Angeles. How many major auto routes are there from Boston to Los Angeles via Detroit? Read More »

A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt are made?

Counting /

Relevant definitions for this exercise: THE PRODUCT RULE: “Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do the second task, then there are

A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt are made? Read More »