To prove the statement (𝑎𝑏𝑎−1)𝑛 = 𝑎𝑏𝑛𝑎−1, we can use the method of mathematical induction.

Base Case (n=1):
(𝑎𝑏𝑎−1)¹ = 𝑎𝑏¹𝑎−1, which is true.

Inductive Step:
Assume the statement is true for some positive integer k, i.e., (𝑎𝑏𝑎−1)ᵏ = 𝑎𝑏ᵏ𝑎−1.

We need to prove that the statement is true for k+1, i.e., (𝑎𝑏𝑎−1)ᵏ⁺¹ = 𝑎𝑏ᵏ⁺¹𝑎−1.

(𝑎𝑏𝑎−1)ᵏ⁺¹ = (𝑎𝑏𝑎−1)ᵏ * (𝑎𝑏𝑎−1) = 𝑎𝑏ᵏ𝑎−1 * 𝑎𝑏𝑎−1 (by the inductive hypothesis)

= 𝑎𝑏ᵏ(𝑎−1𝑎)𝑏𝑎−1 = 𝑎𝑏ᵏ𝑏𝑎−1 = 𝑎𝑏ᵏ⁺¹𝑎−1.

Hence, the statement is true for all positive integers n.

Question

To prove the statement (𝑎𝑏𝑎−1)𝑛 = 𝑎𝑏𝑛𝑎−1, we can use the method of mathematical induction.

Base Case (n=1): 
(𝑎𝑏𝑎−1)¹ = 𝑎𝑏¹𝑎−1, which is true.

Inductive Step:
Assume the statement is true for some positive integer k, i.e., (𝑎𝑏𝑎−1)ᵏ = 𝑎𝑏ᵏ𝑎−1.

We need to prove that the statement is true for k+1, i.e., (𝑎𝑏𝑎−1)ᵏ⁺¹ = 𝑎𝑏ᵏ⁺¹𝑎−1.

(𝑎𝑏𝑎−1)ᵏ⁺¹ = (𝑎𝑏𝑎−1)ᵏ * (𝑎𝑏𝑎−1) = 𝑎𝑏ᵏ𝑎−1 * 𝑎𝑏𝑎−1 (by the inductive hypothesis)

= 𝑎𝑏ᵏ(𝑎−1𝑎)𝑏𝑎−1 = 𝑎𝑏ᵏ𝑏𝑎−1 = 𝑎𝑏ᵏ⁺¹𝑎−1.

Hence, the statement is true for all positive integers n.

Knowee AI · Accepted Answer

To prove the statement (𝑎𝑏𝑎−1)𝑛 = 𝑎𝑏𝑛𝑎−1, we can use the method of mathematical induction.

Base Case (n=1): 
(𝑎𝑏𝑎−1)¹ = 𝑎𝑏¹𝑎−1, which is true.

Inductive Step:
Assume the statement is true for some positive integer k, i.e., (𝑎𝑏𝑎−1)ᵏ = 𝑎𝑏ᵏ𝑎−1.

We need to prove that the statement is true for k+1, i.e., (𝑎𝑏𝑎−1)ᵏ⁺¹ = 𝑎𝑏ᵏ⁺¹𝑎−1.

(𝑎𝑏𝑎−1)ᵏ⁺¹ = (𝑎𝑏𝑎−1)ᵏ * (𝑎𝑏𝑎−1) = 𝑎𝑏ᵏ𝑎−1 * 𝑎𝑏𝑎−1 (by the inductive hypothesis)

= 𝑎𝑏ᵏ(𝑎−1𝑎)𝑏𝑎−1 = 𝑎𝑏ᵏ𝑏𝑎−1 = 𝑎𝑏ᵏ⁺¹𝑎−1.

Hence, the statement is true for all positive integers n.

. Let G be a group and 𝑎, 𝑏 ∈ 𝐺. Then prove that (𝑎𝑏𝑎−1)𝑛 = 𝑎𝑏𝑛𝑎−1