MAM - Indices, Exp, AP, GP

The recurrence rule for sequence A is Tn+1 = Tn - 5, where T2 = 11.

Sequence B has a rule of the form Tn+1 = kTn where T1 = a and k = -2.

a) For sequence A state the first four terms, the shape of the graph of Tn against n and the equation for the graph.

b) For sequence B, describe the sequence and the associated graph.

Solve each of the following exactly.
a)  $x^{-1}=2$  
b)  $x^{\frac{1}{3}}=2$  

c)  $2^{3x}=16$  

Consider the sequence 100, 93, 86, ……

a) Determine:

(i) the recursive definition for the sequence.

(ii) the general term of the sequence, Tn.

b) How many terms of the sequence are positive?

c) Is 23 a term of this sequence?

An English student is writing an essay. He writes 128 words in the first hour, and for each successive hour he writes 1.25 times as much has he has written in the previous hour.

a) How many words did he write during the third hour?

b) Write an explicit definition for Wn = the number of words written during the nth hour.

c) Write a recursive definition for Wn.

A five metre post is hit into the ground by a pile driver.

The first hit drives the post in one metre, the second a further 50 centimetres, the third a further 25 cm, and so on. The amounts driven in form a geometric sequence.

a) State the value of r.

b) How far would the post be driven into the ground if the pile driver could continue hitting forever?

c) What percentage of the total possible distance is reached after:

ten hits?

d) n hits?

Soup is heated in the microwave oven, then removed and allowed to cool. Its temperature T, in °C, x minutes after being removed, is given by the formula:

T = 140(0.85)^x

a) What was the temperature of the soup when it was taken from the oven?

b) At what % rate is the temperature decreasing?

c) Predict the temperature of the soup after 5 minutes.

d) Soup is drinkable at a temperature of 20° C. Determine, to the nearest 0.1 minute, when this will happen.

e) Another food item is placed in the microwave, heated, and left to cool. It cools at a fixed % rate, until its temperature eventually remains relatively constant at k °C. Comment on the value of k.

a) A sequence is defined recursively by the rule

Tn+2 = 3 x Tn+1 - n x Tn

T1 = 9 and T2 = 5

Determine the value of T19.

b) Esme is a marathon runner and her usual training run is 5 km each day. She has decided to gradually increase the distance she runs each day.

On the first day of her new training schedule she runs 5 km and then increases the distance by an extra 800 metres each day.

How far did she run on the fifth day of her new schedule?

c) How far in total has she run after 5 days of her new schedule?

d) Esme’s coach has determined, in the lead up to her next race, that she should not run more than

15 km on any one day and 175 km in total on this new schedule.

On which day of the new schedule does she stop adding on the extra 800 metres?

e) How many days will the new schedule last?

The average rate of inflation from the beginning of 2008 to the beginning of 2018 in Australia has been 1.95% per year.

a) The cost of buying 2L of milk cost $2.75 at the beginning of 2008. How much did it cost a year later?

b) If this average rate of inflation remains constant for the foreseeable future, write a suitable function that models the yearly price of a 2L bottle of milk, since the beginning of 2008.

c) If the model from part (b) remains valid, by the beginning of which year would you expect to pay double the 2008 price?

In 1990, Cathy earned $20 000 and spent all of it in that year. In the subsequent 20 years, her salary increased by 6% per annum while inflation caused her expenditure to increase by $1 000 per annum.

a) Determine a recursive rule for her

(i) earnings in the nth year.

(ii) expenditure in the nth year.

b) By the end of which year would Cathy have earned a total of $799 854.53?

c) Calculate how much Cathy saved by the end of 2010.

A radioactive substance is decaying exponentially. The amount of the substance, M in grams, is related to the time, t in years, relative to a fixed starting time. The relationship is M₁ = 500e^-0.003t.

a) How much of the substance existed originally?

b) What is the half-life of this substance?

c) What is the instantaneous rate of change of M1 when t = 70?

d) Another substance decays according to the rule M₂ = 600(0.996)^t where M₂ and t have the same units, and the same time origin as for M₁.

Which substance is decaying more quickly? Explain.