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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 2020 HSC 12

Calculate the sum of the arithmetic series  `4 + 10 + 16 + … + 1354`.  (3 marks)

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`153\ 454`

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`a = 4, \ l = 1354, \ d = 10 – 4 = 6`

`text(Find)\ n:`

`T_n` `= a + (n + 1)d`
`1354` `= 4 + (n – 1)6`
`1354` `= 6n – 2`
`n` `= 1356/6`
  `= 226`

 

`:. S_226` `= n/2 (a + l)`
  `= 226/2(4 + 1354)`
  `= 153\ 454`

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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  1. `d = 3`
  2. `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 2017 HSC 12c

In an arithmetic series, the fifth term is 200 and the sum of the first four terms is 1200.

Find the value of the tenth term.  (3 marks)

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`0`

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`T_n = a + (n – 1) d`

`=>a + 4d = 200 …\ (1)`

 

`S_n = n/2 [2a + (n – 1) d]`

`=>4a + 6d = 1200 …\ (2)`

 

`text(Multiply)\ (1) xx 4`

`=>4a + 16d = 800 …\ (1 prime)`

 

`text(Subtract)\ \ (1 prime) – (2)`

`10d` `= -400`
`d` `= -40`

 

`text(Substitute)\ \ d = -40\ \ text(into)\ (1)`

`a – 160` `= 200`
`a` `= 360`
`:. T_10` `= 360 – 9(-40)`
  `= 0`

Financial Maths, 2ADV M1 SM-Bank 5 MC

The first three terms of an arithmetic sequence are  `1, 3, 5 . . .`

The sum of the first  `n`  terms of this sequence, `S_n`, is

  1. `S_n = n^2`
  2. `S_n = 2n`
  3. `S_n = 2n - 1`
  4. `S_n = 2n + 1`
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`A`

Show Worked Solution

`text(Sequence is  1, 3, 5 , . . .)`

`text(AP where)\ \ \ a` `= 1, and`
`d` `= 3 – 1 = 2`
`S_n` `= n / 2 [ 2a + (n – 1) d ]`
  `= n / 2 [ 2 xx 1 + (n – 1) 2 ]`
  `= n / 2 [ 2 + 2n – 2 ]`
  `= n^2`

 
`=> A`

Financial Maths, 2ADV M1 2008 HSC 1f

Find the sum of the first 21 terms of the arithmetic series  3 + 7 + 11 + ...   (2 marks)

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`903`

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`S` `= 3 + 7 + 11 + …`
`a` `= 3`
`d` `= 7 – 3 = 4`

 

`:. S_21` `= n/2 [2a + (n – 1) d]`
  `= 21/2 [2 xx 3 + (21 – 1)4]`
  `= 21/2 [6 + 80]`
  `= 903`

Financial Maths, 2ADV M1 2014 HSC 12a

Evaluate the arithmetic series  2 + 5 + 8 + 11 + ... + 1094.    (2 marks) 

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`200\ 020`

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`2 + 5 + 8 + … + 1094`

`AP\ \ text(where)\ \ a = 2,\ \ \ d = 5-2 = 3`
 

`text(Find)\ n:`

`T_n` `= a + (n\ – 1) d`
`1094` `= 2 + (n\ – 1)3`
`3n\ – 3` `= 1092`
`3n` `= 1095`
`n` `= 365`

 

`:. S_365` `= n/2 (a + l)`
  `= 365/2 (2 + 1094)`
  `= 200\ 020`