Proof by Induction (Edexcel International A Level (IAL) Further Maths: Further Pure 1): Exam Questions

Prove by induction that for

where is a constant.

The transformation represented by matrix M followed by the transformation represented by matrix N is represented by the matrix B

(i) Determine N in the form where , , and are integers.

[1]

(ii) Determine B

[2]

Hexagon is transformed onto hexagon by matrix B

Given that the area of is 720 square units, determine the area of

Prove by induction that for

is divisible by 57

Prove by induction that for

Prove by induction that for

is divisible by 7

Prove, by induction, that for ,

is divisible by 18

Prove by induction that for all positive integers

A sequence of numbers is defined by

Prove by induction that, for

Prove by induction that, for

is divisible by 16

Prove by induction that, for

Using the standard summation formulae, show that

where , and are constants to be determined.

Determine the value of for which

A sequence of numbers is defined by

Prove by induction that, for

Prove by induction that, for all positive integers ,

is divisible by 11

Prove by induction that for

Hence show that

where , and are integers to be found.

Using your answers to part (b), find the value of

Prove by induction that is divisible by 21 for all positive integers .

A sequence of numbers , , ,....is defined by

Prove by induction that, for

Prove by induction that, for , is a multiple of 7.

(i) Prove by induction that, for

[6]

(ii) Prove by induction that, for

is divisible by 7.

[6]

Prove by induction that, for

is divisible by 7.