Discrete Math: Sequences and Induction

The value of a new car in 2015 was $40,000. It depreciates 7% each year. How much will the car be worth in 2024?

John bought a new home in 2002. The value of the home increases 4% each year. If the price of the house is $225,000 in 2015, how much did he pay for it in 2002?

A sample contains 100 counts of bacteria. The bacteria triples every 15 minutes. How much bacteria will there be in 1 hour?

Prove that for any integer $n > 4$, $n! > 2^n$.

Prove that $n! > 2^n$ for $n \geq 4$ using mathematical induction.

Prove that $1 + 2 + 3 + \ldots + n = \frac{n(n+1)}{2}$ using mathematical induction.

Show that $1 + 2 + \ldots + n = \frac{n(n + 1)}{2}$.

Prove by induction that the sum of the series $\sum_{r=1}^{n} r^2$ is equal to $\frac{n(n+1)(2n+1)}{6}$.

Prove that the sum of the squares of the first $n$ natural numbers can be expressed as: $1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$ for all natural numbers $n \geq 1$ using proof by induction.

Define the nth number $a_n$ in a Fibonacci sequence such that $a_n = a_{n-1} + a_{n-2}$ for $n \geq 2$, with initial conditions $a_0 = 0$ and $a_1 = 1$.

For the sequence defined by $0, 2, 6, 12, 20, 30, 42$, show that $a_n - a_{n-1} = 2n$ and prove that $a_n = n(n+1)$ given $a_0 = 0$.

Find the sum of $n^2$ from $n = 1$ to $n = 5$.

Find the sum of $2^n$ from $n = 1$ to $n = 6$.

Find the sum of $3n + 2$ from $n = 1$ to $n = 5$.

Find the sum of the first 100 terms of the sequence given by $4n + 5$.

Calculate the sum of the first four terms of the sequence given by $\frac{3}{2}^{n-1}$.

Determine the sum of the infinite geometric sequence represented by $8 \left( \frac{2}{3} \right)^{n-1}$.