Question

Prove by induction: The sum of the cubes of any three consecutive natural numbers is divisible by $9.$

Short answer

The sum of the cubes of any three consecutive natural numbers is divisible by 9.

Step-by-step solution

Base Case

To prove by induction, we first verify the base case. Let's check whether the sum of the cubes of the first three consecutive natural numbers, 1, 2, and 3, is divisible by 9.Calculate:$$1^3+2^3+3^3=1+8+27=36$$Since 36 is divisible by 9, the base case holds true.

Inductive Hypothesis

Assume that for some natural number $k$, the sum of the cubes of three consecutive numbers $k,k+1,$ and $k+2$ is divisible by 9. That means:$$k^3+(k+1{)}^3+(k+2{)}^3=9m$$Where $m$ is an integer.

Inductive Step

Prove that the sum of the cubes of the next three consecutive numbers, $k+1,k+2,$ and $k+3$, is also divisible by 9.Calculate the sum:$$(k+1{)}^3+(k+2{)}^3+(k+3{)}^3$$Using the binomial theorem, expand:$$=(k+1{)}^3+(k+2{)}^3+(k+3{)}^3$$Simplifying:$$k^3+3k^2+3k+1+k^3+6k^2+12k+8+k^3+9k^2+27k+27$$Combine like terms:$$3k^3+18k^2+42k+36$$Factor out 9:$$=9(k^3+2k^2+7k+4)$$This shows the expression is divisible by 9.

Conclusion

Since both the base case holds and the inductive step is complete, by mathematical induction, the sum of the cubes of any three consecutive natural numbers is divisible by 9.

Key concepts

Natural Numbers

Natural numbers are the numbers that we use for counting. They start from 1 and go on indefinitely: 1, 2, 3, 4, etc. Unlike whole numbers, they do not include zero. Natural numbers are intuitive as they represent real-world quantities like objects or things that we count. Natural numbers can be added, subtracted, multiplied, and divided (except by 0) to produce other natural numbers. These numbers have properties such as:

• Closure under addition and multiplication: Adding or multiplying two natural numbers will always give another natural number.
• Associative property: The way in which numbers are grouped in an operation does not change the result (e.g., (1 + 2) + 3 is the same as 1 + (2 + 3)).
• Distributive property: The sum of two numbers, each multiplied by a third number, is equal to the sum of each multiplied individually by the third number (e.g., 2*(3 + 4) is the same as 2*3 + 2*4).

Understanding these properties helps us do advanced operations like mathematical induction.

Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows us to expand expressions raised to a power, particularly those involving two terms (hence bi-nomial). Given ":[(a+b)^n]", we can expand it to a sum of terms using binomial coefficients. The general form of the binomial expansion is: $$(a+b{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})a^{n-k}{b}^k$$ Each term in the expansion has a coefficient $(\genfrac{}{}{0ex}{}{n}{k})$, where $(\genfrac{}{}{0ex}{}{n}{k})$ is a binomial coefficient and is calculated as $$\frac{n!}{k!(n-k)!}$$. The binomial theorem is particularly useful in proving algebraic identities and solving combinatorial problems. When dealing with cubes of numbers, such as those in our problem, the binomial theorem helps us expand expressions into simpler terms that can be added or factored easily.

Factor

A factor is a number that divides another number completely without leaving any remainder. For instance, 3 is a factor of 9 because 9 divided by 3 equals 3, leaving no remainder. Factors can be found for all integers and are fundamental in simplifying expressions and solving equations. To find factors, one can divide the number in question by various integers to see if the result is a whole number.

• 1 and the number itself are always factors of any number.
• Numbers can have multiple factors and knowing them is crucial for division and multiplication problems.

Factoring becomes particularly useful in algebra when we aim to simplify equations or expressions, such as the expression we encounter when proving problems through induction. Factoring allows us to express numbers like 9 as a product ( 3 x 3), simplifying many operations.

Divisibility Rules

Divisibility rules are shortcuts or quick checks to determine if one number divides another completely. These rules ease the process of determining divisibility without performing the actual division. Here are some common divisibility rules:

• A number is divisible by 2 if its last digit is even.
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 9 if the sum of its digits is divisible by 9.

Understanding and using divisibility rules can save a lot of time in math problems. In our exercise, we used the divisibility rule for 9. By confirming that the sum of the cubes of any three consecutive natural numbers gives a result whose digits sum to a number divisible by 9, we proved that the entire expression itself is divisible by 9.