Chapter 10 #10 Certainly! Below is the transcription of the text from the image, formatted for an educational website:---**Mathematical Proofs: Induction and Counterexamples**Explore the fundamental mathematical concept of proving statements using induction and counterexamples. Prove each of the following statements with either mathematical induction, strong induction, or find the smallest counterexample.1. **Arithmetic Series Sum** Prove that: \(1 + 2 + 3 + 4 + \ldots + n = \frac{n^2 + n}{2}\) for every positive integer \(n\).2. **Square Sum Formula** Prove that: \(1^2 + 2^2 + 3^2 + 4^2 + \ldots + n^2 = \frac{n(n+1)(2n+1)}{6}\) for every positive integer \(n\).3. **Cube Sum Formula** Prove that: \(1^3 + 2^3 + 3^3 + 4^3 + \ldots + n^3 = \left(\frac{n(n+1)}{2}\right)^2\) for every positive integer \(n\).4. **Polynomial Summation** If \(n \in \mathbb{N}\), then: \(1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + 4 \cdot 5 + \ldots + n(n+1) = \frac{n(n+1)(n+2)}{3}\).5. **Exponential Equation** If \(n \in \mathbb{N}\), then: \(2^1 + 2^2 + 2^3 + \ldots + 2^n = 2^{n+1} - 2\).6. **Series Expression** Prove that: \(\sum_{i=1}^{n} (8i-5) = 4n^2 - n\) for every positive integer \(n\).7. **Arithmetic Progression** If \(n \in \mathbb{N}\), then: \(1 \cdot 3 + 2 \cdot 4 +

Solution: To show : 3|(52n-1) , for every integer n≥0Note that if a≡b modulo m then m|(a-b)............…(1)clearly, 52≡1 modulo 3 ⇒ (52)n≡ (1)n modulo 3 (........ by the property , if a≡b modulo m then an≡bn modulo m , for every integer n≥0 ) ⇒ 52n ≡ 1 modulo 3 , for every integer n≥0 therefore, from (1) we can write 3|(52n-1) , for every integer n≥0 hence, the proof.