Geometric sequences & series

A sequence has terms u₁, u₂, u₃, … defined by u₁ = 3 and u_n+1 = u_n² - 5 for n ⩾ 1.

Find the values of u₂, u₃ and u₄.

Describe the behaviour of the sequence.

An arithmetic progression has first term a and common difference d, where a and d are non-zero. The first, third and fourth terms of the arithmetic progression are consecutive terms of a geometric progression with common ratio r.

Find d in terms of a.

Find the common ratio of the geometric progression.

The owners of an online shop believe that their sales can be modelled by S = ab^t, where a and b are both positive constants, S is the number of items sold in a month and t is the number of complete months since starting their online shop.

The sales for the first six months are recorded, and the values of log₁₀S are plotted against t in the graph below. The graph is repeated in the Printed Answer Booklet.

Explain why the graph suggests that the given model is appropriate.

The owners believe that a = 120 and b = 1.15 are good estimates for the parameters in the model.

Show that the graph supports these estimates for the parameters.

Use the model S = 120 × 1.15^t to predict the number of items sold in the seventh month after

opening.

Use the model S = 120 × 1.15^t to predict the number of months after opening when the

total number of items sold after opening will first exceed 70 000.

Comment on how reliable this prediction may be.

An artist is creating a design for a large painting. The design includes a set of steps of varying heights. In the painting the lowest step has height 20 cm and the height of each other step is 5% less than the height of the step immediately below it.

In the painting the total height of the steps is 205 cm, correct to the nearest centimetre.

Determine the number of steps in the design.

The positive integers x, y and z are the first, second and third terms, respectively, of an arithmetic progression with common difference - 4.

Show that y satisfies the equation y⁴ - 16y² - 225 = 0.

Hence determine the sum to infinity of the geometric progression.

An arithmetic progression has first term 2 and common difference d, where d ≠ 0. The first, third and thirteenth terms of this progression are also the first, second and third terms, respectively, of a geometric progression.

By determining d, show that the arithmetic progression is an increasing sequence.

In this question you should assume that - 1 < x < 1.

For the binomial expansion of (1 - x)^-2

find and simplify the first four terms,

write down the term in xⁿ.

Write down the sum to infinity of the series 1 + x + x² + x³ + ….

Hence or otherwise find and simplify an expression for 2 + 3x + 4x² + 5x³ + … in the form where a and b are constants to be determined.

Andy and Bev are playing a game.

• The game consists of three points.

• On each point, P(Andy wins) = 0.4 and P(Bev wins) = 0.6.

• If one player wins two consecutive points, then they win the game, otherwise neither player wins.

Determine the probability of the following events.

Andy wins the game.

Neither player wins the game.

Andy and Bev now decide to play a match which consists of a series of games.

• In each game, if a player wins the game then they win the match.

• If neither player wins the game then the players play another game.

Determine the probability that Andy wins the match.

Find the probability that S is odd.

Find the probability that S is greater than 8, given that S is odd.

Sheila sometimes needs several attempts to start her car in the morning. She models the number of attempts she needs by the discrete random variable Y defined as follows. P(Y = y+1) = (Y = y) for all positive integers y.

Find P(Y = 1).

Give a reason why one of the variables, X or Y, might be more appropriate as a model for the number

of attempts that Sheila needs to start her car.