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Worksheet: Explicit and Recursive Formulas of Arithmetic Sequences

Q1:

The fifth term of an arithmetic sequence is 50 and the tenth term is 25 times greater than the second term. Find the general term .

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  • E

Q2:

The fifth term of an arithmetic sequence is and the tenth term is 2 times greater than the second term. Find the general term .

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Q3:

The fifth term of an arithmetic sequence is 210 and the tenth term is 3 times greater than the second term. Find the general term .

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Q4:

Find, in terms of , the general term of an arithmetic sequence whose sixth term is 46 and the sum of the third and tenth term is 102.

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Q5:

Find, in terms of 𝑛 , the general term of an arithmetic sequence whose ninth term is βˆ’ 7 1 7 and sixteenth term is βˆ’ 1 3 4 7 .

  • A π‘Ž = βˆ’ 2 0 6 4 5 𝑛 + 4 8 4 5 𝑛
  • B π‘Ž = 9 3 𝑛 βˆ’ 9 0 𝑛
  • C π‘Ž = 4 8 4 5 𝑛 βˆ’ 2 0 6 4 5 𝑛
  • D π‘Ž = βˆ’ 9 0 𝑛 + 9 3 𝑛
  • E π‘Ž = βˆ’ 8 7 𝑛 βˆ’ 6 3 0 𝑛

Q6:

Find, in terms of , the general term of an arithmetic sequence whose sixth term is and the sum of the third and tenth term is .

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Q7:

The fifth term of an arithmetic sequence is 4 and the tenth term is times greater than the second term. Find the general term .

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Q8:

Find, in terms of , the general term of an arithmetic sequence whose sixth term is 30 and the sum of the third and tenth term is 67.

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Q9:

Find, in terms of 𝑛 , the general term of an arithmetic sequence whose ninth term is 478 and sixteenth term is 891.

  • A π‘Ž = 1 3 6 9 5 𝑛 βˆ’ 3 2 1 6 𝑛
  • B π‘Ž = βˆ’ 5 3 𝑛 + 5 9 𝑛
  • C π‘Ž = βˆ’ 3 2 1 6 𝑛 + 1 3 6 9 5 𝑛
  • D π‘Ž = 5 9 𝑛 βˆ’ 5 3 𝑛
  • E π‘Ž = 6 5 𝑛 + 4 1 3 𝑛

Q10:

The graph represents the triangle wave function 𝑇 ( π‘₯ ) , which is periodic, piecewise linear, and defined for all real numbers.

Let π‘Ž 𝑛 be the 𝑛 th positive solution to the equation 𝑇 ( π‘₯ ) = βˆ’ 1 . Starting from π‘Ž = 3 2 1 , write a recursive formula for π‘Ž 𝑛 .

  • A π‘Ž = π‘Ž + 3 2 𝑛 + 1 𝑛 for 𝑛 β‰₯ 1 . π‘Ž = 3 2 1
  • B π‘Ž = π‘Ž + 1 𝑛 + 1 𝑛 for 𝑛 β‰₯ 1 . π‘Ž = 3 2 1
  • C π‘Ž = π‘Ž + 1 2 𝑛 + 1 𝑛 for 𝑛 β‰₯ 1 . π‘Ž = 3 2 1
  • D π‘Ž = π‘Ž + 2 𝑛 + 1 𝑛 for 𝑛 β‰₯ 1 . π‘Ž = 3 2 1
  • E π‘Ž = π‘Ž + 5 2 𝑛 + 1 𝑛 for 𝑛 β‰₯ 1 . π‘Ž = 3 2 1

What is the set of numbers which satisfy the equation 𝑇 ( π‘₯ ) = 1 ?

  • A  … , βˆ’ 7 2 , βˆ’ 3 2 , 1 2 , 5 2 , … 
  • B  … , βˆ’ 7 2 , βˆ’ 3 2 , 3 2 , 7 2 , … 
  • C  … , βˆ’ 5 2 , βˆ’ 1 2 , 1 2 , 5 2 , … 
  • D { … , βˆ’ 2 , βˆ’ 1 , 1 , 2 , … }
  • E  … , βˆ’ 3 2 , βˆ’ 1 2 , 0 , 1 2 , 5 2 , … 

The part of the graph through the origin ( 0 , 0 ) coincides with the line 𝑦 = 2 π‘₯ . Use this to find one solution to 𝑇 ( π‘₯ ) = 1 2 . Use the symmetries of the graph to find the next positive solution.

  • A π‘₯ = 1 2 . π‘₯ = 9 4
  • B π‘₯ = 1 2 . π‘₯ = 3 4
  • C π‘₯ = 1 4 . π‘₯ = 3 4
  • D π‘₯ = 3 4 . π‘₯ = 9 4
  • E π‘₯ = 1 4 . π‘₯ = 9 4

Find the first two positive solutions to 𝑇 ( π‘₯ ) = βˆ’ 0 . 3 4 6 .

  • A 1 . 1 7 3 , 1 . 8 2 7
  • B 1 . 3 4 6 , 3 . 3 4 6
  • C 1 . 1 7 3 , 3 . 1 7 3
  • D βˆ’ 0 . 1 7 3 , 1 . 1 7 3
  • E 1 . 3 4 6 , 1 . 6 5 4

Find the value of 𝑇 ( 𝑒 ) 5 , giving your answer correct to 3 decimal places.

Q11:

A cable television offers its service at $45 per month and a one-time setup fee of $19.95. Express the total amount paid 𝑃 ( 𝑛 ) after 𝑛 β‰₯ 0 months by a recursive formula.

  • A 𝑃 ( 𝑛 + 1 ) = 1 9 . 9 5 + 4 5 𝑛
  • B 𝑃 ( 𝑛 ) = 1 9 . 9 5 + 4 5 𝑛
  • C 𝑃 ( 𝑛 ) = 𝑃 ( 𝑛 βˆ’ 1 ) + 4 5 , 𝑃 ( 0 ) = 4 5
  • D 𝑃 ( 𝑛 + 1 ) = 𝑃 ( 𝑛 ) + 4 5 , 𝑃 ( 0 ) = 1 9 . 9 5
  • E 𝑃 ( 𝑛 + 1 ) = 𝑃 ( 𝑛 ) + 4 5 𝑛

Q12:

The arithmetic mean between the third and seventh term of a sequence is 36 and the tenth term exceeds the double of the fourth term by 6 Find the general term, , of the arithmetic sequence.

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Q13:

The arithmetic mean between the third and seventh term of a sequence is and the tenth term exceeds the double of the fourth term by 44 Find the general term, , of the arithmetic sequence.

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Q14:

Consider the following growing pattern, shown for 𝑛 = 1 , 𝑛 = 2 , and 𝑛 = 3 .

Write an expression for the number of dots in the 𝑛 th such pattern.

  • A 𝑛 + 1
  • B 2 𝑛 + 1
  • C 𝑛 βˆ’ 1
  • D 2 𝑛 βˆ’ 1
  • E 𝑛 + 2

Q15:

The third term in an arithmetic sequence is 2 and the sixth term is 11. If the first term is π‘Ž 1 , what is an equation for the 𝑛 th term of this sequence?

  • A π‘Ž = 𝑛 βˆ’ 1 𝑛
  • B π‘Ž = 7 𝑛 βˆ’ 2 𝑛
  • C π‘Ž = 4 𝑛 βˆ’ 1 3 𝑛
  • D π‘Ž = 3 𝑛 βˆ’ 7 𝑛
  • E π‘Ž = 𝑛 + 5 𝑛

Q16:

Find the sequence and its general term of all the even numbers greater than 62.

  • A ,
  • B ,
  • C ,
  • D ,

Q17:

Find the general term of the arithmetic sequence which satisfies the relations π‘Ž + π‘Ž = βˆ’ 3 0 6 8 and π‘Ž Γ— π‘Ž = 5 2 5 7 9 .

  • A π‘Ž = 5 5 5 2 𝑛 βˆ’ 5 7 0 𝑛
  • B π‘Ž = βˆ’ 1 0 𝑛 + 4 5 𝑛
  • C π‘Ž = βˆ’ 1 0 𝑛 βˆ’ 2 5 𝑛
  • D π‘Ž = βˆ’ 1 0 𝑛 + 5 5 𝑛
  • E π‘Ž = βˆ’ 1 0 𝑛 + 3 5 𝑛

Q18:

Find the general term of the arithmetic sequence which satisfies the relations π‘Ž + π‘Ž = 1 4 1 2 1 4 and π‘Ž Γ— π‘Ž = 7 1 3 1 5 .

  • A π‘Ž = βˆ’ 7 2 𝑛 + 1 4 𝑛
  • B π‘Ž = βˆ’ 3 𝑛 + 4 3 𝑛
  • C π‘Ž = βˆ’ 3 𝑛 + 4 𝑛
  • D π‘Ž = βˆ’ 3 𝑛 + 4 6 𝑛
  • E π‘Ž = βˆ’ 3 𝑛 + 4 0 𝑛

Q19:

Find, in terms of 𝑛 , the general term of the sequence 3 5 0 + 1 2 , 3 5 0 + 1 3 , 3 5 0 + 1 4 , 3 5 0 + 1 5 , … .

  • A 3 5 0 + ( βˆ’ 1 ) 𝑛 + 1 𝑛
  • B 3 5 0 + ( βˆ’ 1 ) 𝑛 + 1 𝑛 + 1
  • C 3 5 0 + 1 𝑛
  • D 3 5 0 + 1 𝑛 + 1

Q20:

Find, in terms of 𝑛 , the general term of the sequence ( 4 4 , 7 0 , 9 6 , 1 2 2 , … ) .

  • A βˆ’ 2 6 𝑛 + 1 8
  • B 2 6 𝑛 βˆ’ 1 8
  • C 1 8 𝑛 + 2 6
  • D 2 6 𝑛 + 1 8

Q21:

Jennifer started working out to get healthier. She worked out for fourteen minutes on the first day and increased her exercise by six minutes every day. Find, in terms of , the term of the sequence which represents her plan.

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