A sequence \( a_1, a_2, a_3, \ldots \) is defined by letting \( a_1 = 3 \) and \( a_k = 7a_{k-1} \) for all natural numbers \( k \ge 2 \). Show that \( a_n = 3 \cdot 7^{n-1} \) for all natural numbers.
A sequence \( b_0, b_1, b_2, \ldots \) is defined by letting \( b_0 = 5 \) and \( b_k = 4 + b_{k-1} \) for all natural numbers \( k \). Show that \( b_n = 5 + 4n \) for all natural number \( n \) using mathematical induction.
A sequence \( d_1, d_2, d_3, \ldots \) is defined by letting \( d_1 = 2 \) and \( d_k = \dfrac{d_{k-1}}{k} \) for all natural numbers \( k \ge 2 \). Show that \( d_n = \dfrac{2}{n!} \) for all \( n \in \mathbb{N} \).
Prove that for all \( n \in \mathbb{N} \)
\(\cos \alpha + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \cdots + \cos(\alpha + (n-1)\beta) = \dfrac{\cos\left(\alpha + \dfrac{n-1}{2}\beta\right) \sin\left(\dfrac{n\beta}{2}\right)}{\sin\left(\dfrac{\beta}{2}\right)}\).
Prove that \(\cos \theta \cos 2\theta \cos 2^2\theta \cdots \cos 2^{n-1}\theta = \dfrac{\sin 2^n \theta}{2^n \sin \theta}\), for all \( n \in \mathbb{N} \).
Prove that \( \sin \theta + \sin 2\theta + \sin 3\theta + \cdots + \sin n\theta = \dfrac{\sin\left(\dfrac{n\theta}{2}\right) \sin\left(\dfrac{(n+1)\theta}{2}\right)}{\sin\left(\dfrac{\theta}{2}\right)} \), for all \( n \in \mathbb{N} \).
Show that \( \dfrac{n^5}{5} + \dfrac{n^3}{3} + \dfrac{7n}{15} \) is a natural number for all \( n \in \mathbb{N} \).
Prove that \( \dfrac{1}{n+1} + \dfrac{1}{n+2} + \cdots + \dfrac{1}{2n} > \dfrac{13}{24} \), for all natural numbers \( n > 1 \).
Prove that the number of subsets of a set containing \( n \) distinct elements is \( 2^n \), for all \( n \in \mathbb{N} \).