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Upper 6 Pure MathematicsUnit 2 Module 2 Rein forcement Questions Series and Sequences

1. (a) The first termof an arithmetic sequence is 30 and the common difference

is1.5.

(i) Find the value of the 25th term. [2]

The rth term of the sequence is 0.

(ii) Find the value of r. [2]

The sum of the first nterms of the sequence is Sn.

(iii) Find the largest positive value of Sn. [3]

(b) A trading company made a profit of $50000 in 2006 (Year 1).A model for future trading predicts that profits will increase year by yearin a geometric sequence with common ratio r, r> 1.

The model therefore predicts that in 2007 (Year 2) a profit of $50000rwill be made.

(i) Write down an expression for the predicted profit in Year n. [1]

The model predicts that in Year n, the profit made will exceed $200000.

(ii) Show that n>rlog

4log+ 1. [3]

Using the model with r= 1.09,

(iii) find the year in which the profit made will first exceed $200000,

[2](iv) find the total of the profits that will be made by the company over

the 10 years from 2006 to 2015 inclusive, giving your answer tothe nearest $10000.

2. Mr.John Slick takes out an investment with an investment company which requires

making a fixed payment of $A at the beginning of each year. At the end of the

investment period John expects to receive a payout sum of money which is equal to

the total payments made together with interest added at the end of each year at a rate

of r% per annum of the total sum in the fund.

a. Write down expressions for

i. The amounts at the beginning of years 4 and 5 [2]

ii. payout sums at the end of years 4 and 5 [2]

b. Write an expression for the amount at the beginning of the nth year [2]

c. Show that the payout sum in (b) is AR (Rn1) / (R-1) dollars where r>1 [7]

d. Find the value of A to the nearest dollar when n = 20, r = 5 and the payout

sum in (c) above is $500,000 [7]

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3. An oil well produced 100,000 barrels of oil in its first year but output fell by 7.5%

each year thereafter

a. Determine to 2sf the number of barrels of oil that could be produced

i. From the well as the maximum total output [5]

ii. By the well in the first 15 years [4]

iii. What percentage of the total output of oil would be left after 15 years?

[2]

4. a. A car was purchased at the beginning of the year for P dollars. The value of the carat the end of the year is estimated to be the value at the beginning of the yearmultiplied by ( 11/q) where q is an element of N.

i. Copy and complete the table below [3]

Year 1 2 3 4 5Value at thebeginning ofyear $

Value at theend of year $

ii. Describe fully the sequence shown in the table [2]iii. Determine in terms of P and q the value of the car n years after purchase [1]

b. If the original value of the car was $20000 and the value at the end of the 4thyear

was $8192 find:i. The value of q [5]ii. The estimated value of the car after 5 years [2]iii. The least integral value of n the number of years after purchase for which theestimated value of eth car falls below $500 [7]

5. Matthew started a savings account at a local bank by depositing $5 in the first week.In each succeeding week after the first he added twice the amount deposited in theprevious weekDerive an expression fro

a.

The amount deposited in the rth week in terms of r [3]b.

The total amount in the account after n weeks in terms of n [3]Calculate the minimum number of weeks it would take for the amount in the accountto exceed $1000 if no withdrawal is made. [3]

6.

Johns starting annual salary at a company is $25000. His contract at the companystates that his annual salary in subsequent years will be increased by 2% over thesalary of the previous year.Find to the nearest dollar:

a. Johns salary for the tenth year with the company [4]b. The total amount of money which the company would have paid to John at the

end of his first ten years with the company [5]

7. A truck bought for $15000 depreciates at the rate of 12 % each year. Calculate thevalue of eth truck

a. After 1 year [2]b. After r years [2]c. When its value first falls below $5000 [3]