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Financial Maths, 2ADV M1 2021 HSC 14

The first term of an arithmetic sequence is 5. The sum of the first 43 terms is 2021.

What is the common difference of the sequence?   (2 marks)

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`2`

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`T_1 = a = 5`

`S_43` `= n/2 [2a + (n-1)d]`
`2021` `= 43/2 (10 + 42d)`
`2021` `= 215 + 903d`
`903d` `= 1806`
`:. d` `= 2`

Financial Maths, 2ADV M1 2019 HSC 12b

In an arithmetic series, the fourth term is 6 and the sum of the first 16 terms is 120.

Find the common difference.   (3 marks)

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`1/3`

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`T_4 = 6,`

`a + 3d = 6\ …\ \ (1)`

`S_16 = 120,`

`16/2(2a + 15d)` `= 120`
`16a + 120d` `= 120\ …\ \ (2)`

 
`text(Substitute)\ \ a = 6-3d\ \ text{from (1) into (2):}`

`16(6-3d) + 120d` `= 120`
`96-48d + 120d` `= 120`
`72d` `= 24`
`d` `= 1/3`

Financial Maths, 2ADV M1 2018 HSC 11d

In an arithmetic series, the third term is 8 and the twentieth term is 59.

  1. Find the common difference.   (1 mark)

  2. Find the 50th term.   (2 marks)

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i.    `d = 3`

ii.   `149`

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i.   `a + 2d` `= 8 qquad text{… (1)}`
  `a + 19d` `= 59 qquad text{… (2)}`

 
`text(Substract)\ \ (2)-(1)`

`17d` `= 51`
`:. d` `= 3`

 

ii.   `text(Find)\ \ T_50`

`text(Substitute)\ \ d = 3\ \ text{into (1)}`

`=> a = 12`

`T_n` `=a+(n-1)d`
`:. T_50` `= 2 + 49 xx 3`
  `= 149`

Financial Maths, 2ADV M1 EQ-Bank 32

There are 10 checkpoints in a 4500 metre orienteering course. Checkpoint 1 is the start and checkpoint 10 is the finish.

The distance between successive checkpoints increases by 50 metres as each checkpoint is passed.

Calculate the distance, in metres, between checkpoint 2 and checkpoint 3.   (3 marks)

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`text(350 m)`

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`text(9 intervals exist between 10 checkpoints)`

`=>\ text(9 distances form an AP where:)\ d=50, \ S_9=4500`

`S_n` `= n/2[2a + (n-1)d]`
`4500` `= 9/2 [2a + (9-1) × 50]`
`4500` `= 9/2[2a + 400]`
`9a` `= 2700`
`a` `= 300`

  
`text(Distance between checkpoint 1 and 2)`

`= a=300\ text(m)`

`:.\ text(Distance between checkpoint 2 and 3)`

`= a+d=350\ text(m)`