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Financial Maths, 2ADV M1 2025 HSC 13

The numbers, 75, \(p\), \(q\), 2025, form a geometric sequence.

Find the values of \(p\) and \(q\).   (2 marks)

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\(p=225, \ q=675\)

Show Worked Solution

\(a=75, \ 75r=p, \ 75r^2=q, \ 75r^3=2025\)

\(\text{Using}\ \ 75r^3=2025:\)

\(r=\sqrt[3]{\dfrac{2025}{75}}=3\)

\(p=75 \times 3 = 225\)

\(q=75 \times 3^{2}=675\)

Financial Maths, 2ADV M1 2023 HSC 21

The fourth term of a geometric sequence is 48 .

The eighth term of the same sequence is `3/16`.

Find the possible value(s) of the common ratio and the corresponding first term(s).   (3 marks)

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`a=3072,\ r=1/4, or`

`a=-3072,\ r=-1/4`

Show Worked Solution

`T_4=ar^3=48\ …\ (1)`

`T_8=ar^7=3/16\ …\ (2)`

`(ar^7)/(ar^3)` `=(3/16)/48`
`r^4` `=1/256`
`r` `=+-1/4`

 
`text{If}\ \ r=1/4`

`a(1/4)^3` `=48`
`a/64` `=48`
`a` `=3072`

 
`text{If}\ \ r=-1/4,\ \ a=-3072`

`a(-1/4)^3` `=48`
`-a/64` `=48`
`a` `=-3072`
  

`:.\ a=3072,\ r=1/4\  or\  a=-3072,\ r=-1/4`

Financial Maths, 2ADV M1 2017 HSC 16b

A geometric series has first term  `a`  and limiting sum 2.

Find all possible values for  `a`.   (3 marks)

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`0 < a < 4`

Show Worked Solution

`S_oo = a/(1 – r)`

♦ Mean mark 39%.

`text(If)\ \ S_oo = 2:`

`2` `= a/(1-r)`
`2(1-r)` `= a`
`1-r` `= a/2`
`r` `= 1-a/2`

  
`text(S)text(ince)\ \ |r| < 1,`

`|1-a/2| < 1`

`1-a/2` `< 1` `or qquad -(1-a/2)` `< 1`
`a/2` `> 0` `a/2` `< 2`
`a` `> 0` `a` `< 4`

  
`:. 0 < a < 4`

Financial Maths, 2ADV M1 EQ-Bank 6 MC

The first four terms of a geometric sequence are  `6400\ ,\  t_2\  ,\  8100\ , -9112.5`

The value of  `t_2` is

  1. `-7250` 
  2. `-7200`
  3. `-1700`
  4. `7200`
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`B`

Show Worked Solution

`text(GP is)\ \ 6400,  t_2,  8100,  –9112.5`

`r` `=t_2/t_1 = t_3/t_2`
`t_2 / 6400` `= (-9112.5) / 8100`
`t_2` `= (-9112.5 × 6400) / 8100= -7200`

 
`=>  B`

Financial Maths, 2ADV M1 2014 HSC 14d

At the beginning of every 8-hour period, a patient is given 10 mL of a particular drug.

During each of these 8-hour periods, the patient’s body partially breaks down the drug. Only  `1/3`  of the total amount of the drug present in the patient’s body at the beginning of each 8-hour period remains at the end of that period.  

  1. How much of the drug is in the patient’s body immediately after the second dose is given?   (1 mark)

  2. Show that the total amount of the drug in the patient’s body never exceeds 15 mL.   (2 marks)

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a.    `13.33\ text{mL  (2 d.p.)}`

b.    `text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

a.    `text(Let)\ \ A =\ text(Amount of drug in body)`

`text(Initially)\ A = 10`

`text(After 8 hours)\ \ \ A` `=1/3 xx 10`
`text(After 2nd dose)\ \ A` `= 10 + 1/3 xx 10\ text(mL)`
  `=13.33\ text{mL  (2 d.p.)}`

 

b.    `text(After the 3rd dose)`

`A_3` `= 10 + 1/3 (10 + 1/3 xx 10)`
  `= 10 + 1/3 xx 10 + (1/3)^2 xx 10`

 
`  =>\ text(GP where)\ a = 10,\ r = 1/3`

`text(S)text(ince)\ \ |\ r\ | < 1:`

`S_oo` `= a/(1\ – r)`
  `= 10/(1\ – 1/3)`
  `= 10/(2/3)`
  `= 15`

 

  
`:.\ text(The amount of the drug will never exceed 15 mL.)`

Financial Maths, 2ADV M1 2008 HSC 5b

Consider the geometric series

`5+10x+20x^2+40x^3+\ ...`

  1. For what values of `x` does this series have a limiting sum?   (2 marks)

  2. The limiting sum of this series is `100`.
  3. Find the value of `x`.   (2 marks)
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a.    `-1/2<x<1/2`

b.    `19/40`

Show Worked Solutions

a.    `text(Limiting sum when)\ |\ r\ |<1`

`r=T_2/T_1=(10x)/5=2x`

`:.\ |\ 2x\ |<1`

`text(If)\ \ 2x` `>0` `text(If)\ \ 2x` `<0`
`2x` `<1` `-(2x)` `<1`
`x` `<1/2` `2x` `> -1`
    `x` `> -1/2`

  
`:. text(Limiting sum when)\ \ -1/2<x<1/2`
  

b.    `text(Given)\  S_oo=100,  text(find) \ x`

`=> S_oo=a/(1-r)=100`

` 5/(1-2x)` `=100`
`100(1-2x)` `=5`
`200x` `=95`
`:.\ x` `=95/200=19/40`