Mixed AP and GP

1. A series is log(x^-1) + log x + log (x³) + log (x⁵) + ...

Is the series arithmetic or geometric? Fully justify your answer.

2. The numbers a, b and 9 are in arithmetic progression while the numbers a, b and 12 are in geometric progression.

Find the two sequences.

Answer.The APs could be either
3, 6, 9 or 27, 18, 9.
The corresponding GPs could be
3, 6, 12 or 27, 18, 12.

3. The numbers x, y and x + y are in arithmetic progression while the numbers
x, y and 20 are in geometric progression.

(i) Show that there is only one pair of numbers which provides a solution for these two series.

(ii) Use that pair of numbers to write out the first three terms of each of the two series.

Answer.The only pair is x = 5 and y = 10.
The AP is 5, 10 and 15.
The GP is 5, 10, 20.

4. With water conservation of great importance with global warming becoming more of a problem, two CSIRO scientists have developed a special device to count the number of raindrops falling each minute over a specified area during a storm.

They establish that the number of raindrops R_t (to the nearest whole drop) falling in the t^th minute forms a series whose t^th term can be estimated from the formula

(i) What is the initial number of raindrops falling?

(ii) Show that the model predicts that after 10 minutes, there will be about 107 raindrops falling per minute.

(iii) Find the total number of raindrops which fell over the designated area throughout the first half hour of a storm.

(iv) Describe the pattern of the rainfall throughout the half hour.

Answer.(i) R₀ = 4.
(iii) Total = 106,387 raindrops.

5. The terms in the sequence m, mr, mr², mr³, ... clearly form a geometric sequence.

Show that the logarithms of these terms form an arithmetic sequence.

6. The first three terms in a sequence are 3, a, b and 192.

Find the values of a and b if the sequence is:

(i) arithmetic.

(ii) geometric.

Answer(i) a = 66, b = 129.
(ii) a = 12, b = 48.

8. An arithmetic progression has 2 as its first term. The first, fourth and tenth terms of that progression are in a geometric sequence.

Find the first three terms in the arithmetic progression.

Answer2, 8/3 (2.67) and 10/3 (3.33).

9. A golf tournament is organized and the prize money for the first 14 place winners is distributed as follows:

• First Prize = $500,000
• The next 4 placed players each receive 70% of the prize received by the previous player (for example 4th placed receives 70% of third prize and so on);
• The next 9 placed players each receive $10,000 less than the previous
prize winner.
• There are no tied place winners.

Find:

(i) the value of the 4th prize;
(ii) the value of the 14th prize;
(iii) The total amount of prize money available in the tournament is set at $2 million. The balance between that total and the prize money awarded to the first 14 players is allocated to the first person to shoot a hole in one during the round. How much is the reward for the hole in one?

Note: Don't just write down all the prizes - use your understanding of series.

Answer(i) Prize for 4th is $171,500.
(ii) Prize for 14th is $30,050.
(iii) Hole in one prize is $3,000.

10. Nathan and Damien have been offered jobs in a factory making flour. The jobs are for a fixed period of 20 days during school holidays.

Nathan agrees to be paid 5 cents for the first day of work, 10c for the second day, 20c for the third day with the company doubling his wage each day of the 20 day contract.

Damien agrees to be paid $500 for the first day of work with the company increasing his wage by $50 per day for each day of the 20 day contract.

(i) How much will Nathan earn on the 7th day of his work?

(ii) How much will Damien earn on the 7th day of his work?

(iii) Damien gets a better offer and only works 10 days. Calculate the total amount Damien had earned during those 10 days.

(iv) What is the minimum number of days Nathan has to work to equal Damien's wages?

Answer(i) Nathan earned $3.20.
(ii) Damien earned $800.
(iii) Damien earned $7,250.
(iv) Nathan had to work 18 days.