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CORE*, FUR1 2006 VCAA 8 MC

Paula started a stamp collection. She decided to buy a number of new stamps every week.

The number of stamps bought in the `n`th week, `t_n`, is defined by the difference equation

`t_n = t_(n-1) + t_(n-2)\ \ \ text(where)\ \ \ t_1 = 1 and t_2 = 2`

The total number of stamps in her collection after five weeks is

A.     `8`

B.   `12`

C.   `15`

D.   `19`

E.   `24` 

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`D`

Show Worked Solution

`t_1=1,\ \ \ t_2=2\ \ \ text{(given)}`

♦♦ Mean mark 33%.
STRATEGY: Always ask if the question wants a term or a sum of terms. This will minimise a very common mistake in this topic area.
`:. t_3` `=t_2 + t_1 = 2+1=3`
`t_4` `=3+2=5`
`t_5` `=5+3= 8`

 

`:.\ text(Total stamps after 5 weeks)`

`=1+2+3+5+8=19`

`rArr D`

CORE*, FUR1 2011 VCAA 5 MC

The difference equation

`t_(n+2) = t_(n+1) + t_n`  where  `t_1 = a`  and  `t_2 = 7`

generates a sequence with  `t_5 = 27`.

The value of  `a`  is

A.   0

B.   1

C.   2

D.   3

E.   4

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`D`

Show Worked Solution

`t_(n+2) = t_(n+1) + t_n\ \ text(where)\ \ t_1 = a\ \ text(and)\ \ t_2 = 7`

`text(Calculating this equation from)\ \ n = 1,`

`t_3 ` ` = t_2 + t_1`
  ` = 7 + a`
`t_4 ` ` = t_3 + t_2`
  ` = 7 + a + 7`
  ` = 14 + a`
`t_5` ` = t_4 + t_3`
`:. 27 ` ` = 14 + a + 7 + a`
`2a ` ` = 6`
`a ` ` = 3`

 
`=> D`

CORE*, FUR1 2014 VCAA 7 MC

The first term of a Fibonacci-related sequence is  `p`.

The second term of the same Fibonacci-related sequence is  `q`.

The difference in value between the fourth and fifth terms of this sequence is

A.   `p - q`  

B.   `q - p`  

C.   `p + q`

D.   `p + 2q`

E.   `2p + 3q`

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`C`

Show Worked Solution

`text(Fibonacci sequence general form is)`

♦ Mean mark 42%.

`t_(n+2) = t_(n+1) + t_n`

`t_1 = p`

`t_2 = q`

`t_3 = p + q`

`t_4 = (p + q) + q = p + 2q`

`t_5 = p + 2q + (p + q) = 2p + 3q`

`∴ t_5 – t_4` `= 2p + 3q – (p + 2q)`
  `= p + q`

 
`=>  C`

CORE*, FUR1 2009 VCAA 1 MC

The first six terms of a Fibonacci related sequence are shown below.

4,  7,  11,  18,  29,  47, ...

The next term in the sequence is

A.     58

B.     65

C.     76

D.     94

E.   123

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`C`

Show Worked Solution

`text(Fibonacci general term,)`

`T_(n+2) = T_(n+1) + T_n`

`:.\ text(Next term)` `= 29 + 47` 
  `= 76`

 
`rArr C`