justinburr.net

Mathematical Induction

1 Prove that ${\color{White} 1^2+2^2+3^2+...+n^2=\frac{n(n+1)(2n+1)}{6}}$ for all ${\color{White} n\geq1}$

2 Prove that ${\color{White} 1^3+2^3+3^3+...+n^3=(1+2+3+...+n)^2}$ for all ${\color{White} n\geq1}$

3 Prove that ${\color{White} x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1})}$ for all ${\color{White} n\geq1}$

4 Prove that ${\color{White} 1*2+2*3+3*4+...+n(n+1)=\frac{n(n+1)(n+2)}{3}}$ for all ${\color{White} n\geq1}$

5 Prove that ${\color{White} 1+3+5+...+(2n-1)=n^2}$ for all ${\color{White} n\geq1}$

6 Prove that ${\color{White} \frac{1}{2*1}+\frac{1}{2*3}+\frac{1}{3*4}+...+\frac{1}{n(n+1)}=\frac{n}{n+1}}$ for all ${\color{White} n\geq1}$

7 Prove that ${\color{White} F_{1}+F_{2}+F_{3}+...+F_{n}=F_{n+2}-1}$ for all ${\color{White} n\geq1}$

8 Prove that ${\color{White} F_{1}+F_{3}+F_{5}+...+F_{2n-1}=F_{2n}}$ for all ${\color{White} n\geq1}$

9 Prove that ${\color{White} F_{2}+F_{4}+F_{6}+...+F_{2n}=F_{2n+1}-1}$ for all ${\color{White} n\geq1}$

10 Prove that ${\color{White} F_{n+1}^{2}-F_{n}F_{n+2}=(-1)^n}$ for all ${\color{White} n\geq1}$

11 Prove that ${\color{White} F_{1}F_{2}+F_{2}F_{3}+F_{3}F_{4}+...+F_{2n-1}F_{2n}=F_{2n}^{2}}$ for all ${\color{White} n\geq1}$

12 Prove that ${\color{White} F_{1}F_{2}+F_{2}F_{3}+F_{3}F_{4}+...+F_{2n}F_{2n+1}=F_{2n+1}^{2}-1}$ for all ${\color{White} n\geq1}$

13 Prove that ${\color{White} L_{n}=L_{n-1}+L_{n-2}}$ for all ${\color{White} n\geq3}$

14 Prove that ${\color{White} F_{2n}=F_{n}L_{n}}$ for all ${\color{White} n\geq1}$

15 Prove that ${\color{White} L_{1}+2L_{2}+4L_{3}+8L_{4}+...+2^{n-1}L_{n}=2^{n}F_{n+1}-1}$ for all ${\color{White} n\geq1}$

16 Prove that ${\color{White} n(n^{2}-1)(3n+2)}$ is divisible by 24 for all ${\color{White} n\geq1}$