Proof by induction

Given

prove by mathematical induction that, for n

Given

describe fully the single geometrical transformation P represented by the matrix B

The transformation Q is represented by the matrix Aⁿ

The transformation P followed by the transformation Q is the transformation R, which is represented by the matrix C

Determine C in terms of n

Given that, for a particular value of n, the transformation R maps the point with coordinates (27, 1) to the point with coordinates (a, a), where a is a constant,

determine the matrix that represents the transformation Q

Prove by induction that, for all positive integers n,

Prove by induction that, for all positive integers n,

Prove by mathematical induction that, for n

Triangle T has vertices A, B and C.

Triangle T is transformed to triangle T′ by the transformation represented by Mⁿ where n

Given that

• triangle T has an area of 5 cm²

• triangle T′ has an area of 1215 cm²

• vertex A (2, –2) is transformed to vertex A ′(123, –2)

determine

the value of n

the value of a

Prove by mathematical induction that, for n

Triangle T has vertices A, B and C.

Triangle T is transformed to triangle T′ by the transformation represented by Mⁿ where n

Given that

• triangle T has an area of 5 cm²

• triangle T′ has an area of 1215 cm²

• vertex A (2, –2) is transformed to vertex A ′(123, –2)

determine

the value of n

the value of a

There are three lily pads on a pond. A frog hops repeatedly from one lily pad to another.

The frog starts on lily pad A, as shown in Figure 1.

In a model, the frog hops from its position on one lily pad to either of the other two lily pads with equal probability.

Let p_n be the probability that the frog is on lily pad A after n hops.

Explain, with reference to the model, why p₁ = 0

The probability p_n satisfies the recurrence relation

Prove by induction that, for n 1

Use the result in part (b) to explain why, in the long term, the probability that the

A recurrence system is defined by

u_n+2 = 9(n + 1)² u_n − 3u_n+1 n 1

u₁ = −3, u₂ = 18

Prove by induction that, for n ,

Prove by induction that for n ⁺

Prove by induction that for all positive odd integers n

is divisible by 15

Prove by induction that for n ⁺

Prove by induction that for all positive odd integers n

is divisible by 15

Prove by induction that for all positive integers n

is divisible by 5

Prove by induction that for all positive integers n

is divisible by 5