F.4 M2 MI 1條題 15分

When n = 1 ,
LHS = 2^(1-1) = 1
RHS = 2²⁻³ + 2¹⁻² = 0.5 + 0.5 = 1
The statement is true for n = 1.
Assume the statement is true for some positive integer n = k
i.e.
1 + 2 + 3 + 4 + ... + 2ᵏ⁻¹ = 2²ᵏ⁻³ + 2ᵏ⁻² When n = k+1 ,
L.H.S.
1 + 2 + 3 + 4 + ... + 2ᵏ
= 1 + 2 + 3 + 4 + ... +2ᵏ⁻¹ + (2ᵏ⁻¹ + 1) + (2ᵏ⁻¹ + 2) + ... + (2ᵏ⁻¹ + 2ᵏ⁻¹)
= 2²ᵏ⁻³ + 2ᵏ⁻² + (2ᵏ⁻¹ + 2ᵏ⁻¹ + ... + 2ᵏ⁻¹) + 1 + 2 + 3 + 4 + ... + 2ᵏ⁻¹
= 2²ᵏ⁻³ + 2ᵏ⁻² + 2ᵏ⁻¹(2ᵏ⁻¹) + 2²ᵏ⁻³ + 2ᵏ⁻²
= 2 (2²ᵏ⁻³ + 2ᵏ⁻²) + 2ᵏ⁻¹(2ᵏ⁻¹)
= 2²ᵏ⁻² + 2ᵏ⁻¹ + 2²ᵏ⁻²
= 2²ᵏ⁻¹ + 2ᵏ⁻¹
= 2²⁽ᵏ⁺¹⁾⁻³ + 2⁽ᵏ⁺¹⁾⁻² The statement is also true for n = k+1 when it is true for n = k. By the principle of mathematical induction, it is true for all positive integers n.