Worksheet: Recursive Formula of a Sequence

In this worksheet, we will practice finding the recursive formula of a sequence.

Q1:

Find a recursive formula for the sequence 486,162,54,18,6,2,23.

  • A 𝑎 = 1 3
  • B 𝑎 = 1 3 𝑎
  • C 𝑎 = 1 2
  • D 𝑎 = 1 3
  • E 𝑎 = 1 2

Q2:

Write a recursive formula for the following sequence that is valid for 𝑛2:1,2,3,4,5,

  • A 𝑎 = 𝑎 1
  • B 𝑎 = 𝑎
  • C 𝑎 = 𝑎
  • D 𝑎 = 𝑎 1
  • E 𝑎 = 𝑎 + 1

Q3:

Using the absolute value function, write a recursive formula for the following sequence that is valid for 𝑛2:1,2,3,4,5,

  • A 𝑎 = ( 1 ) ( | 𝑎 | + 1 )
  • B 𝑎 = ( 1 ) ( | 𝑎 | 1 )
  • C 𝑎 = ( 1 ) ( | 𝑎 | 1 )
  • D 𝑎 = ( 1 ) ( | 𝑎 | + 1 )
  • E 𝑎 = ( 1 ) ( | 𝑎 | + 1 )

Q4:

Find a recursive formula for the sequence 𝑎=5+1.

  • A 𝑎 = 5 𝑎 4 , 𝑎 = 6
  • B 𝑎 = 5 𝑎 6 , 𝑎 = 6
  • C 𝑎 = 5 𝑎 + 4 , 𝑎 = 6
  • D 𝑎 = 5 𝑎 + 6 , 𝑎 = 6
  • E 𝑎 = 5 𝑎 + 4 , 𝑎 = 6

Find a recursive formula for the sequence 𝑎=2(5)+1.

  • A 𝑎 = 5 𝑎 + 4 , 𝑎 = 1 1
  • B 𝑎 = 5 𝑎 + 6 , 𝑎 = 1 1
  • C 𝑎 = 5 𝑎 6 , 𝑎 = 1 1
  • D 𝑎 = 5 𝑎 + 4 , 𝑎 = 1 1
  • E 𝑎 = 5 𝑎 4 , 𝑎 = 1 1

Find a recursive formula for the sequence 𝑎=13(5)+1.

  • A 𝑎 = 5 𝑎 6 , 𝑎 = 8 3
  • B 𝑎 = 5 𝑎 + 6 , 𝑎 = 8 3
  • C 𝑎 = 5 𝑎 4 , 𝑎 = 8 3
  • D 𝑎 = 5 𝑎 + 4 , 𝑎 = 8 3
  • E 𝑎 = 4 𝑎 + 5 , 𝑎 = 8 3

Find a recursive formula for the sequence 𝑎=5𝑎+1 and 𝑎=376.

  • A 𝑎 = 3 . 0 8 ( 5 ) + 4
  • B 𝑎 = 2 . 9 3 6 ( 5 ) + 1
  • C 𝑎 = 3 ( 5 ) + 4
  • D 𝑎 = 3 . 0 0 8 ( 5 ) + 1
  • E 𝑎 = 3 ( 5 ) + 1

Q5:

Find the first five terms of the sequence with general term 𝑎=𝑎+5, where 𝑛1 and 𝑎=13.

  • A ( 1 8 , 2 3 , 2 8 , 3 3 , 3 8 )
  • B ( 1 3 , 1 8 , 2 3 , 2 8 , 3 3 )
  • C ( 8 , 3 , 2 , 7 , 1 2 )
  • D ( 1 3 , 8 , 3 , 2 , 7 )

Q6:

The 𝑛th term in a sequence is given by 𝑎=𝑎+𝑎. Find the first six terms of this sequence, given that 𝑎=0 and 𝑎=1.

  • A ( 1 , 2 , 3 , 5 , 8 , 1 3 )
  • B ( 0 , 1 , 2 , 3 , 5 , 8 )
  • C ( 0 , 1 , 1 , 2 , 3 , 5 )
  • D ( 0 , 1 , 1 , 2 , 3 , 4 )

Q7:

The 𝑛th term in a sequence is given by 𝑎=𝑛𝑎. Find the first six terms of this sequence, given that 𝑎=118.

  • A ( 1 1 8 , 2 3 6 , 3 5 4 , 4 7 2 , 5 9 0 , 7 0 8 )
  • B ( 1 1 8 , 1 1 8 , 2 3 6 , 7 0 8 , 2 , 8 3 2 , 1 4 , 1 6 0 )
  • C ( 2 3 6 , 3 5 4 , 4 7 2 , 5 9 0 , 7 0 8 , 8 2 6 )
  • D ( 1 1 8 , 2 3 6 , 7 0 8 , 2 , 8 3 2 , 1 4 , 1 6 0 , 8 4 , 9 6 0 )

Q8:

Given the sequence defined by 𝑎=𝑎+𝑛𝑥, where 𝑎=27 and 𝑎=78, find the value of 𝑥.

Q9:

Find 𝑎+𝑎+𝑎 given 𝑎=3 and 𝑎=𝑎+58.

  • A 9 2 1 8
  • B 1 2 3 8
  • C 6 9 4
  • D 2 6 7 8

Q10:

Given that 𝑎=8 and that 𝑎=12𝑎 for 𝑛1, find a formula for 𝑎 in terms of 𝑛.

  • A 𝑎 = 2
  • B 𝑎 = 8 2
  • C 𝑎 = 1 2
  • D 𝑎 = 2
  • E 𝑎 = 2

Q11:

Find the arithmetic sequence in which 𝑎=100 and 𝑎=4𝑎.

  • A ( 1 0 0 , 4 0 0 , 5 0 0 , )
  • B ( 1 0 0 , 3 0 0 , 4 0 0 , )
  • C ( 1 0 0 , 3 0 0 , 5 0 0 , )
  • D ( 1 0 0 , 2 0 0 , 3 0 0 , )

Q12:

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

List the values of 𝑇(0), 𝑇(1), and 𝑇(1,234).

  • A0, 1, 1
  • B1, 1, 0
  • C1, 1, 1
  • D0, 0, 0
  • E0, 1, 1

List the values of 𝑇12, 𝑇32, 𝑇52, and 𝑇1,2332.

  • A1, 1, 1, 1
  • B1, 1, 1, 1
  • C 1 , 1 , 1, 1
  • D1, 1, 1, 1
  • E1, 1, 1, 1

What is 𝑇4,9332?

If we are given that 𝑇(𝑏) is negative, what can we conclude about the number 𝑏?

  • AThere is some integer 𝑛 for which 2𝑛<𝑏<2𝑛+1.
  • B 𝑏 is an even integer.
  • CThere is some integer 𝑛 for which 2𝑛+1<𝑏<2𝑛+2.
  • D 𝑏 is an odd integer.

Find the equation of the line segment on which the point (𝜋,𝑇(𝜋)) lies.

  • A 𝑦 = 2 ( 3 𝑥 1 )
  • B 𝑦 = 1 2 ( 𝑥 1 2 )
  • C 𝑦 = 2 ( 𝑥 3 )
  • D 𝑦 = 2 ( 𝑥 + 3 )
  • E 𝑦 = 4 ( 𝑥 3 )

Hence find the value of 𝑇(𝜋) correct to 3 decimal places.

Q13:

Find, in terms of 𝑛 the general term of the sequence which satisfies the relation 𝑎=22𝑎, where 𝑛1 and 𝑎=22.

  • A 2 2 𝑛
  • B ( 2 2 )
  • C 2 2 𝑛
  • D ( 2 2 )

Q14:

Consider the sequence given by 𝑓(0)=0,𝑓(𝑛+1)=1𝑓(𝑛).

List the numbers at positions 2, 3, and 4.

  • A0, 1, 0
  • B1, 0, 1
  • C1, 1, 0
  • D0, 0, 1
  • E0, 1, 1

What is the number at position 12,341?

What is the range of this sequence?

  • A { 1 , 2 }
  • B { 0 , 1 }
  • C { 0 , 1 , 2 }
  • D { 2 , 3 , 4 }

Q15:

Find the relationship between the terms in the sequence (26,26,52,78,130,208,338,).

  • A 𝑎 = 𝑎 + 𝑎
  • B 𝑎 = 𝑎 + 𝑎
  • C 𝑎 = 𝑎 + 𝑎
  • D 𝑎 = 𝑎 + 𝑎

Q16:

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

Let 𝑎 be the 𝑛th positive solution to the equation 𝑇(𝑥)=1. Starting from 𝑎=32, write a recursive formula for 𝑎.

  • A 𝑎 = 𝑎 + 1 for 𝑛1. 𝑎=32
  • B 𝑎 = 𝑎 + 5 2 for 𝑛1. 𝑎=32
  • C 𝑎 = 𝑎 + 2 for 𝑛1. 𝑎=32
  • D 𝑎 = 𝑎 + 1 2 for 𝑛1. 𝑎=32
  • E 𝑎 = 𝑎 + 3 2 for 𝑛1. 𝑎=32

What is the set of numbers which satisfy the equation 𝑇(𝑥)=1?

  • A , 3 2 , 1 2 , 0 , 1 2 , 5 2 ,
  • B , 7 2 , 3 2 , 3 2 , 7 2 ,
  • C , 7 2 , 3 2 , 1 2 , 5 2 ,
  • D { , 2 , 1 , 1 , 2 , }
  • E , 5 2 , 1 2 , 1 2 , 5 2 ,

The part of the graph through the origin (0,0) coincides with the line 𝑦=2𝑥. Use this to find one solution to 𝑇(𝑥)=12. Use the symmetries of the graph to find the next positive solution.

  • A 𝑥 = 3 4 . 𝑥 = 9 4
  • B 𝑥 = 1 4 . 𝑥 = 9 4
  • C 𝑥 = 1 2 . 𝑥 = 3 4
  • D 𝑥 = 1 4 . 𝑥 = 3 4
  • E 𝑥 = 1 2 . 𝑥 = 9 4

Find the first two positive solutions to 𝑇(𝑥)=0.346.

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

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

Q17:

The sequence 1,1,1,5,17 is given by a recursive formula of the form 𝑎=𝑆𝑎+𝑇𝑎 for 𝑛1. Determine this formula.

  • A 𝑎 = 2 𝑎 3 𝑎
  • B 𝑎 = 4 𝑎 + 3 𝑎
  • C 𝑎 = 𝑎 + 2 𝑎
  • D 𝑎 = 3 𝑎 + 2 𝑎
  • E 𝑎 = 5 𝑎 3 𝑎

Q18:

Find the first six terms of the sequence satisfying the relation 𝑎=𝑎+𝑎, where 𝑛1, 𝑎=22, and 𝑎=15.

  • A ( 2 2 , 1 5 , 7 , 8 , 1 , 9 )
  • B ( 3 7 , 5 2 , 8 9 , 1 4 1 , 2 3 0 , 3 7 1 )
  • C ( 2 2 , 1 5 , 3 7 , 5 2 , 8 9 , 1 4 1 )
  • D ( 2 2 , 1 5 , 3 7 , 5 2 , 8 9 , 1 4 1 )

Q19:

The function 𝑓 is defined by 𝑓(1)=1 and 𝑓(𝑛+1)=2𝑓(𝑛)3 for 𝑛1. Complete the table.

𝑛 1 2 3 4
𝑓 ( 𝑛 )
  • A 1 , 5 , 1 3 , 2 9
  • B 1 , 1 , 1 3 , 5
  • C 1 , 1 , 5 , 1 3
  • D 1 , 1 , 5 , 1 3
  • E 1 , 1 , 1 , 3

Q20:

Find 𝑎 given 𝑎=14𝑛𝑎, 𝑛1, and 𝑎=28.

Q21:

Find the first five terms of the sequence 𝑎, given 𝑎=1𝑎, 𝑛1, and 𝑎=499.

  • A 1 4 9 9 , 4 9 9 , 1 4 9 9 , 4 9 9 , 1 4 9 9
  • B 1 4 9 9 , 4 9 9 , 1 4 9 9 , 4 9 9 , 1 4 9 9
  • C 4 9 9 , 1 4 9 9 , 4 9 9 , 1 4 9 9 , 4 9 9
  • D 4 9 9 , 1 4 9 9 , 4 9 9 , 1 4 9 9 , 4 9 9

Q22:

Consider the sequence 𝑎=𝑝3+𝑞.

Find a recursive formula for this sequence.

  • A 𝑎 = 3 𝑎 + 𝑞
  • B 𝑎 = 3 𝑎 2 𝑞
  • C 𝑎 = 3 𝑎 + 𝑞 + 3 𝑝
  • D 𝑎 = 3 𝑎 2 𝑞 + 3 𝑝

The recursive formula 𝑎=3𝑎+5, where 𝑎=1, can be made explicit with a formula of the form 𝑎=𝑝3+𝑞 for suitable numbers 𝑝 and 𝑞. Find the values of 𝑝 and 𝑞.

  • A 𝑝 = 7 6 and 𝑞=52
  • B 𝑝 = 7 6 and 𝑞=52
  • C 𝑝 = 3 5 and 𝑞=52
  • D 𝑝 = 7 6 and 𝑞=5
  • E 𝑝 = 5 2 and 𝑞=76

Q23:

A sequence is defined by the recursive formula 𝑎=3𝑎2,𝑎=2.

Find the first six terms of this sequence.

  • A 2 , 8 , 2 6 , 8 0 , 2 4 2 , 7 2 8
  • B 2 , 8 , 2 6 , 8 0 , 2 4 2 , 7 2 6
  • C 2 , 8 , 2 7 , 8 0 , 2 4 2 , 7 2 8
  • D 2 , 6 , 1 8 , 7 8 , 2 3 6 , 7 1 0
  • E 2 , 8 , 2 6 , 8 0 , 2 4 0 , 7 2 2

Is this sequence arithmetic, geometric, both, or neither?

  • AGeometric
  • BArithmetic
  • CBoth geometric and arithmetic
  • DNeither geometric nor arithmetic

Find an explicit formula for 𝑏, where 𝑏=𝑎1.

  • A 𝑏 = ( 1 ) 3
  • B 𝑏 = ( 3 )
  • C 𝑏 = 3 𝑛
  • D 𝑏 = 3
  • E 𝑏 = ( 1 ) 3

Using your answer to the previous part, find an explicit formula for 𝑎.

  • A 𝑎 = 1 + 3
  • B 𝑎 = ( 3 ) + 1
  • C 𝑎 = 1 3
  • D 𝑎 = 1 3
  • E 𝑎 = 3 𝑛 + 1

Consider the sequence defined by the recursive formula 𝑐=7𝑐+4𝑐=2,. Find the value of 𝑘 for which 𝑑=𝑐+𝑘 is a geometric sequence with common ratio 7.

  • A 𝑘 = 2 3
  • B 𝑘 = 4 7
  • C 𝑘 = 4
  • D 𝑘 = 3 2
  • E 𝑘 = 4

Using your answer to the previous part, derive an explicit formula for 𝑐.

  • A 𝑐 = 4 3 7 + 2 3
  • B 𝑐 = 8 3 7 2 3
  • C 𝑐 = 7 2 3
  • D 𝑐 = 7 5
  • E 𝑐 = 7 4

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