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.

  • A𝑎=13
  • B𝑎=13𝑎
  • C𝑎=12
  • D𝑎=13
  • E𝑎=12

Q2:

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

  • A𝑎=𝑎1
  • B𝑎=𝑎
  • C𝑎=𝑎
  • D𝑎=𝑎1
  • E𝑎=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,

  • A𝑎=(1)(|𝑎|+1)
  • B𝑎=(1)(|𝑎|1)
  • C𝑎=(1)(|𝑎|1)
  • D𝑎=(1)(|𝑎|+1)
  • E𝑎=(1)(|𝑎|+1)

Q4:

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

  • A𝑎=5𝑎4, 𝑎=6
  • B𝑎=5𝑎6, 𝑎=6
  • C𝑎=5𝑎+4, 𝑎=6
  • D𝑎=5𝑎+6, 𝑎=6
  • E𝑎=5𝑎+4, 𝑎=6

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

  • A𝑎=5𝑎+4, 𝑎=11
  • B𝑎=5𝑎+6, 𝑎=11
  • C𝑎=5𝑎6, 𝑎=11
  • D𝑎=5𝑎+4, 𝑎=11
  • E𝑎=5𝑎4, 𝑎=11

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

  • A𝑎=5𝑎6, 𝑎=83
  • B𝑎=5𝑎+6, 𝑎=83
  • C𝑎=5𝑎4, 𝑎=83
  • D𝑎=5𝑎+4, 𝑎=83
  • E𝑎=4𝑎+5, 𝑎=83

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

  • A𝑎=3.08(5)+4
  • B𝑎=2.936(5)+1
  • C𝑎=3(5)+4
  • D𝑎=3.008(5)+1
  • E𝑎=3(5)+1

Q5:

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

  • A(18,23,28,33,38)
  • B(13,18,23,28,33)
  • C(8,3,2,7,12)
  • D(13,8,3,2,7)

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(1,2,3,5,8,13)
  • 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(118,236,354,472,590,708)
  • B(118,118,236,708,2,832,14,160)
  • C(236,354,472,590,708,826)
  • D(118,236,708,2,832,14,160,84,960)

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.

  • A9218
  • B1238
  • C694
  • D2678

Q10:

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

  • A𝑎=2
  • B𝑎=82
  • C𝑎=12
  • D𝑎=2
  • E𝑎=2

Q11:

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

  • A(100,400,500,)
  • B(100,300,400,)
  • C(100,300,500,)
  • D(100,200,300,)

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
  • C1, 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.
  • B𝑏 is an even integer.
  • CThere is some integer 𝑛 for which 2𝑛+1<𝑏<2𝑛+2.
  • D𝑏 is an odd integer.

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

  • A𝑦=2(3𝑥1)
  • B𝑦=12(𝑥12)
  • C𝑦=2(𝑥3)
  • D𝑦=2(𝑥+3)
  • E𝑦=4(𝑥3)

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.

  • A22𝑛
  • B(22)
  • C22𝑛
  • D(22)

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,).

  • A𝑎=𝑎+𝑎
  • B𝑎=𝑎+𝑎
  • C𝑎=𝑎+𝑎
  • D𝑎=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.

  • A𝑎=𝑎+1 for 𝑛1, 𝑎=32
  • B𝑎=𝑎+52 for 𝑛1, 𝑎=32
  • C𝑎=𝑎+2 for 𝑛1, 𝑎=32
  • D𝑎=𝑎+12 for 𝑛1, 𝑎=32
  • E𝑎=𝑎+32 for 𝑛1, 𝑎=32

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

  • A,32,12,0,12,52,
  • B,72,32,32,72,
  • C,72,32,12,52,
  • D{,2,1,1,2,}
  • E,52,12,12,52,

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.

  • A𝑥=34, 𝑥=94
  • B𝑥=14, 𝑥=94
  • C𝑥=12, 𝑥=34
  • D𝑥=14, 𝑥=34
  • E𝑥=12, 𝑥=94

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

  • A1.346,1.654
  • B1.173,1.827
  • C1.346,3.346
  • D0.173,1.173
  • E1.173,3.173

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.

  • A𝑎=2𝑎3𝑎
  • B𝑎=4𝑎+3𝑎
  • C𝑎=𝑎+2𝑎
  • D𝑎=3𝑎+2𝑎
  • E𝑎=5𝑎3𝑎

Q18:

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

  • A(22,15,7,8,1,9)
  • B(37,52,89,141,230,371)
  • C(22,15,37,52,89,141)
  • D(22,15,37,52,89,141)

Q19:

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

𝑛1234
𝑓(𝑛)
  • A1,5,13,29
  • B1,1,13,5
  • C1,1,5,13
  • D1,1,5,13
  • E1,1,1,3

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.

  • A1499,499,1499,499,1499
  • B1499,499,1499,499,1499
  • C499,1499,499,1499,499
  • D499,1499,499,1499,499

Q22:

Consider the sequence 𝑎=𝑝3+𝑞.

Find a recursive formula for this sequence.

  • A𝑎=3𝑎+𝑞
  • B𝑎=3𝑎2𝑞
  • C𝑎=3𝑎+𝑞+3𝑝
  • D𝑎=3𝑎2𝑞+3𝑝

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.

  • A𝑝=76 and 𝑞=52
  • B𝑝=76 and 𝑞=52
  • C𝑝=35 and 𝑞=52
  • D𝑝=76 and 𝑞=5
  • E𝑝=52 and 𝑞=76

Q23:

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

Find the first six terms of this sequence.

  • A2, 8, 26, 80, 242, 728
  • B2, 8, 26, 80, 242, 726
  • C2, 8, 27, 80, 242, 728
  • D2, 6, 18, 78, 236, 710
  • E2, 8, 26, 80, 240, 722

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.

  • A𝑏=(1)3
  • B𝑏=(3)
  • C𝑏=3𝑛
  • D𝑏=3
  • E𝑏=(1)3

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

  • A𝑎=1+3
  • B𝑎=(3)+1
  • C𝑎=13
  • D𝑎=13
  • E𝑎=3𝑛+1

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

  • A𝑘=23
  • B𝑘=47
  • C𝑘=4
  • D𝑘=32
  • E𝑘=4

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

  • A𝑐=437+23
  • B𝑐=83723
  • C𝑐=723
  • D𝑐=75
  • E𝑐=74

Q24:

Find the first four terms in the sequence 𝑎=12𝑎4. Take 𝑎=36, where 𝑛1 (𝑛 an integer).

  • A𝑎=40, 𝑎=36, 𝑎=32, 𝑎=28
  • B𝑎=80, 𝑎=36, 𝑎=14, 𝑎=3
  • C𝑎=72, 𝑎=36, 𝑎=18, 𝑎=9
  • D𝑎=64, 𝑎=36, 𝑎=22, 𝑎=15
  • E𝑎=20, 𝑎=36, 𝑎=68, 𝑎=132

Q25:

Write down a recursive formula for the sequence 3,9,21,45,91,.

  • A𝑎=2𝑎3, 𝑎=3, 𝑛1
  • B𝑎=𝑎+6, 𝑎=3, 𝑛1
  • C𝑎=2𝑎+3, 𝑎=3, 𝑛1
  • D𝑎=3𝑎, 𝑎=3, 𝑛1
  • E𝑎=3𝑎+6, 𝑎=3, 𝑛1

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