Induction

Author: SolveSheep

Prove formulas just using mathematical induction, one of the most important things in number theory




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1.5 Number Theory Inequalities
Prove Bernoulli's Inequality by Induction

Prove Bernoulli's Inequality by mathematical induction:
If \(x\) is a real number such that \(x \ge -1\), then for every non-negative integer \(n\), the inequality \((1+x)^n \ge 1+nx\) holds.
1.5 Number Theory
Prove the Formula for Triangular Numbers

Use mathematical induction to prove the formula for the nth triangular number: $$T_n = \frac{n(n+1)}{2}$$.
1.5 Number Theory
Prove the Formula for the Sum of Cubes

Use mathematical induction to prove the formula for the sum of the first n cubes: $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$.
1.5 Algebra Number Theory
Prove the Formula for the Sum of Powers of Two by Induction

Use the principle of mathematical induction to prove that for any non-negative integer \( n \), the sum of the first \( n+1 \) powers of two is given by the formula:
$$ \sum_{k=0}^{n} 2^k = 2^{n+1} - 1 $$
1.5 Number Theory
Prove the Formula for the Sum of Squares

Use mathematical induction to prove the formula for the sum of the first n squares: $$1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$$.
1.5 Algebra Number Theory
Prove the Formula for the Sum of an Arithmetic Series by Induction

Use mathematical induction to prove the formula for the sum of the first \( n \) terms of an arithmetic series, which is given by
$$ S_n = \sum_{k=1}^n (a + (k-1)d) = \frac{n}{2}(2a + (n-1)d) $$
for all positive integers \( n \ge 1 \), where \( a \) is the first term and \( d \) is the common difference.
1.5 Algebra Number Theory
Prove the Sum of \(k \cdot k!\) Formula by Induction

Prove by mathematical induction that for all positive integers \(n\), the following formula holds:ç$$ \sum_{k=1}^{n} k \cdot k! = (n+1)! - 1 $$
2.0 Number Theory
Prove a Divisibility Rule by Induction

Use the principle of mathematical induction to prove that for every positive integer \(n \geq 1\), the expression
$$
3^{2n+2} - 8n - 9
$$
is divisible by 64.
2.0 Number Theory Combinatorics
Prove the Formula for the Sum of Fibonacci Numbers by Induction

Let \( F_k \) be the \( k \)-th Fibonacci number defined by \( F_1 = 1 \), \( F_2 = 1 \), and \( F_k = F_{k-1} + F_{k-2} \) for \( k \geq 3 \).
Prove by mathematical induction that for all positive integers \( n \):
$$ \sum_{i=1}^n F_i = F_{n+2} - 1 $$