Write a recursive definition including all steps basis induc

Write a recursive definition (including all steps, basis, inductive hypothesis, etc) for the set {2, 4, 6, ...} of positive even integers. Please give all steps.

Solution

Set : { 2,4,6,8, .... n, n+2 }

Set of positive even numbers is given .

Basis = Initial term = t ₁ = 2

Next term = t ₂ = 4

which can be written as -

t ₂ = 4 = 2+2 = t ₁ + 2

t ₃ = 6

which can be written as -

t ₃ = 6 = 4+2 = t ₂ + 2

similarly, t ₄ = 8 = 6+2 = t ₃ + 2

so, we can see the pattern & find the recursive relation that -

for any n -

t _n = t _n-1 + 2 ; n = 2,3,4,5,.....