Proof by induction

Question 1

inductive series (they give you two statements, Un+1 and Un), inductive step

Answer 1

manipulate Un+1 to get a Uk value then sub in Uk rearrange to get what you would have if you subbed n = k+1 into Un

Question 2

Deductive series, inductive step

Answer 2

put k+1 in sum, sum equals sum to k + expression inside the sum

Question 3

Matrices, inductive step

Answer 3

A^k+1 = A^k * A

Question 4

polynomials divisibility, inductive step

Answer 4

find and simplify f(k+1), take out divisibility factor and f(k)

Question 5

indicies divisibility, inductive step

Answer 5

find f(k+1), manipulate into form f(k) +n

Question 6

inequalities, inductive step

Answer 6

manipulate one side using n=k+1 until its a multiple of n=k
multiply the other side by this multiple
manipulate this side and compare it to when n=k+1
remove middle term leaving n=k+1 expressions

Question 7

standard conclusion

Answer 7

so if the result is true when n=k it has been show to be true for n=k+1
since it is true when n=1 then by induction is is true for all values greater than or equal to 1

Question 8

which roots are always the same

Answer 8

the sum a+b+c = -b/a