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Financial Maths, 2ADV M1 2018 HSC 14d

An artist posted a song online. Each day there were  `2^n + n`  downloads, where `n` is the number of days after the song was posted.

  1. Find the number of downloads on each of the first 3 days after the song was posted.   (1 mark)

  2. What is the total number of times the song was downloaded in the first 20 days after it was posted?   (2 marks)

Show Answers Only

i.    `text(Day 1) : 3`

`text(Day 2) : 6`

`text(Day 3) : 11`

ii.   `2\ 097\ 360`

Show Worked Solution

i.    `text(Day 1:)\ \ 2^1 + 1 = 3`

`text(Day 2:)\ \ 2^2 + 2 = 6`

`text(Day 3:)\ \ 2^3 + 3 = 11`

 

ii.   `text{Total downloads (20 days)}`

♦ Mean mark (ii) 38%.

`= 2^1 + 1 + 2^2 + 2 + … + 2^20 + 20`

`= underbrace(2^1 + 2^2 + … + 2^20)_{text(GP),\ a = 2,\ r=2} + underbrace(1 + 2 + … + 20)_{text(AP),\ a = 1,\ d = 1}`

`= (2(2^20-1))/(2-1) + 20/2(1 + 20)`

`= 2\ 097\ 150 + 210= 2\ 097\ 360`

Financial Maths, 2ADV M1 2005 HSC 7a

Anne and Kay are employed by an accounting firm.

Anne accepts employment with an initial annual salary of  $50 000. In each of the following years her annual salary is increased by $2500.

Kay accepts employment with an initial annual salary of  $50 000. In each of the following years her annual salary is increased by 4%.

  1. What is Anne’s annual salary in her thirteenth year?   (2 marks)

  2. What is Kay’s annual salary in her thirteenth year?   (2 marks)

  3. By what amount does the total amount paid to Kay in her first twenty years exceed that paid to Anne in her first twenty years?   (3 marks)

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a.    `$80\ 000`

b.    `text{$80 052 (nearest dollar)}`

c.    `text{$13 904 (nearest $)}`

Show Worked Solution

a.    `text(Let)\ T_n = text(Anne’s salary in year)\ n`

`T_1 = a = $50\ 000`

`T_2 = a + d = $52\ 500`

`⇒\ text(AP where)\ a = $50\ 000,\ \ d = $2500`

`T_n = a + (n-1)d`

`T_13` `= 50\ 000 + (13-1) xx 2500`
  `=80\ 000`

  
`:.\ text(Anne’s salary in her 13th year is $80 000.)`
  

b.    `text(Let)\ K_1 =text(Kay’s salary in year)\ n`

`K_1` `= a` `= 50\ 000`
`K_2` `= ar` `= 50\ 000 xx 1.04 = 52\ 000`
`⇒\ text(GP where)\ \ a = 50\ 000, \ \ r = 1.04`
`K_n` `= ar^(n-1)`
`K_13` `= 50\ 000 xx (1.04)^12`
  `= $80\ 051.61…`
  `= $80\ 052\ \ \ text{(nearest dollar)}`

  
c.    `text(Anne)`

`S_n` `= n/2[2a + (n-1)d]`
 `S_20` `= 20/2[2 xx 50\ 000 + (20-1)2500]`
  `= 10[100\ 000 + 47\ 500]`
  `= $1\ 475\ 000`

  
`text(Kay)`

`S_n` `= (a(r^n-1))/(r-1)`
`S_20`  `= (50\ 000(1.04^20-1))/(1.04-1)`
  `= $1\ 488\ 903.929…`

   
`text(Difference)`

`= 1\ 488\ 903.929…-1\ 475\ 000`

`= $13\ 903.928…`

`= $13\ 904\ \ \  text{(nearest $)}`
  

`:.\ text(Kay’s total salary exceeds Anne’s by)\ $13\ 904`

Financial Maths, 2ADV M1 2013 HSC 12c

Kim and Alex start jobs at the beginning of the same year. Kim's annual salary in the first year is `$30 000` and increases by 5% at the beginning of each subsequent year. Alex's annual salary in the first year is `$33 000`, and increases by $1500 at the beginning of each subsequent year.

  1. Show that in the 10th year, Kim's annual salary is higher than Alex's annual salary.   (2 marks)

  2. In the first 10 years how much, in total, does Kim earn?   (2 marks)

  3. Every year, Alex saves `1/3` of her annual salary. How many years does it take her to save $87,500?   (3 marks)

Show Answers Only

a.    `text{Proof (See Worked Solutions)}`

b.    `$377\ 336.78`

c.    `text(7  years)`

Show Worked Solutions

a.    `text(Let)\ \ K_n=text(Kim’s salary in Year)\  n`

`{:{:(K_1=a=30\ 000),(K_2=ar^1=30\ 000(1.05^1)):}}{:(\ =>\ GP),(\ \ \ \ \ \ a=30\ 000),(\ \ \ \ \ \ r=1.05):}`

`vdots`

`:.K_10=ar^9=30\ 000(1.05)^9=$46\ 539.85`
  

`text(Let)\ \ A_n=text(Alex’s salary in Year)\ n`

`{:{:(A_1=a=33\ 000),(A_2=33\ 000+1500=34\ 500):}}{:(\ =>\ AP),(\ \ \ \ \ \ a=33\ 000),(\ \ \ \ \ \ d=1500):}`

`vdots`

`A_10=a+9d=33\ 000+1500(9)=$46\ 500`

`=>K_10>A_10`
 

`:.\ text(Kim earns more than Alex in the 10th year)`

  
b.    `text(In the first 10 years, Kim earns)`

`K_1+K_2+\  ….+ K_10`

`S_10` `=a((r^n-1)/(r-1))`
  `=30\ 000((1.05^10-1)/(1.05-1))`
  `=377\ 336.78`

    
`:.\ text(In the first 10 years, Kim earns $377 336.78)`
  

c.    `text(Let)\ T_n=text(Alex’s savings in Year)\ n`

`{:{:(T_1=a=1/3(33\ 000)=11\ 000),(T_2=a+d=1/3(34\ 500)=11\ 500),(T_3=a+2d=1/3(36\ 000)=12\ 000):}}{:(\ =>\ AP),(\ \ \ \ a=11\ 000),(\ \ \ \ d=500):}`
 

`text(Find)\ n\ text(such that)\ S_n=87\ 500`

♦ Mean mark (c) 45%.
IMPORTANT: Using the AP sum formula to create and then solve a quadratic in `n` is challenging and often examined. Students need to solve and interpret the solutions.
`S_n` `=n/2[2a+(n-1)d]`
`87\ 500` `=n/2[22\ 000+(n-1)500]`
`87\ 500` `=n/2[21\ 500+500n]`
`250n^2+10\ 750n-87\ 500` `=0`
`n^2+43n-350` `=0`
`(n-7)(n+50)` `=0`

 
`:.n=7,\ \ \ \ n>0`

`:.\ text(Alex’s savings will be $87,500 after 7 years).`