Worksheet: Recursive Sequences

In this worksheet, we will practice working with sequences in recursive form.

Q1:

Find the first five terms of the sequence with general term 𝑇 = 𝑇 + 5 𝑛 + 1 𝑛 , where 𝑛 1 and 𝑇 = 1 3 1 .

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

Q2:

The 𝑛 t h term in a sequence is given by 𝑇 = 𝑇 + 𝑇 𝑛 + 2 𝑛 + 1 𝑛 . Find the first six terms of this sequence, given that 𝑇 = 0 1 and 𝑇 = 1 2 .

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

Q3:

Find the arithmetic sequence in which 𝑇 = 1 0 0 1 and 𝑇 = 4 𝑇 4 𝑛 𝑛 .

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

Q4:

Given that and that for , find a formula for in terms of .

  • A
  • B
  • C
  • D
  • E

Q5:

Find 𝑇 + 𝑇 + 𝑇 1 3 1 4 1 5 given 𝑇 = 3 1 and 𝑇 = 𝑇 + 5 8 𝑛 𝑛 1 .

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

Q6:

The 𝑛 t h term in a sequence is given by 𝑇 = 𝑛 𝑇 𝑛 + 1 𝑛 . Find the first six terms of this sequence, given that 𝑇 = 1 1 8 1 .

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

Q7:

Find, in terms of 𝑛 the general term of the sequence which satisfies the relation 𝑇 = 2 2 𝑇 𝑛 + 1 𝑛 , where 𝑛 1 and 𝑇 = 2 2 1 .

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

Q8:

The sequence 𝑎 , where 𝑛 1 , is given by

List the next 6 terms 𝑎 , , 𝑎 .

  • A 𝑎 = 6 , 𝑎 = 6 , 𝑎 = 7 , 𝑎 = 7 , 𝑎 = 7 , 𝑎 = 8
  • B 𝑎 = 5 , 𝑎 = 6 , 𝑎 = 6 , 𝑎 = 7 , 𝑎 = 7 , 𝑎 = 8
  • C 𝑎 = 6 , 𝑎 = 6 , 𝑎 = 6 , 𝑎 = 7 , 𝑎 = 7 , 𝑎 = 8
  • D 𝑎 = 5 , 𝑎 = 6 , 𝑎 = 6 , 𝑎 = 7 , 𝑎 = 7 , 𝑎 = 8
  • E 𝑎 = 5 , 𝑎 = 6 , 𝑎 = 6 , 𝑎 = 7 , 𝑎 = 7 , 𝑎 = 8

By listing the elements 𝑎 , 𝑎 , 𝑎 , 𝑎 , , give a formula for 𝑎 , in terms of 𝑛 , for 𝑛 1 .

  • A 𝑎 = 2 ( 𝑛 1 )
  • B 𝑎 = ( 𝑛 + 1 )
  • C 𝑎 = 2 ( 𝑛 + 1 )
  • D 𝑎 = ( 𝑛 1 )
  • E 𝑎 = ( 2 𝑛 1 )

Give a formula for 𝑎 , in terms of 𝑛 , for 𝑛 1 .

  • A 𝑎 = 𝑛 + 1
  • B 𝑎 = 2 𝑛 + 1
  • C 𝑎 = 2 𝑛 1
  • D 𝑎 = 2 ( 𝑛 1 )
  • E 𝑎 = 2 ( 𝑛 + 1 )

Give a formula for 𝑎 , in terms of 𝑛 , for 𝑛 1 .

  • A 𝑎 = 1 2 𝑛
  • B 𝑎 = 2 + 𝑛
  • C 𝑎 = 2 𝑛
  • D 𝑎 = 1 + 2 𝑛
  • E 𝑎 = 1 𝑛

Give a formula for 𝑎 , in terms of 𝑛 , for 𝑛 1 .

  • A 𝑎 = 1 + 2 𝑛
  • B 𝑎 = 2 𝑛
  • C 𝑎 = 1 2 𝑛
  • D 𝑎 = 2 𝑛
  • E 𝑎 = 2 𝑛

What is 𝑎 ?

  • A 𝑎 = 6 1 7 2
  • B 𝑎 = 6 7 1 0
  • C 𝑎 = 6 1 7 0
  • D 𝑎 = 6 1 7 0
  • E 𝑎 = 6 1 7 2

Solve 𝑎 = 1 7 for 𝑛 .

  • A 𝑛 = 3 2
  • B 𝑛 = 3 4
  • C 𝑛 = 3 5
  • D 𝑛 = 3 7
  • E 𝑛 = 3 3

What is the range of the function 𝑎 ?

  • Athe set of negative integers
  • Bthe set of negative rationals
  • Cthe set of postive rationals
  • Dthe set of all integers
  • Ethe set of positive integers

Q9:

Consider the following sequence of dots.

What is the function 𝑓 such that 𝑓 ( 𝑛 ) is the number of dots in the 𝑛 th pattern?

  • A 𝑓 ( 𝑛 ) = 𝑛 ( 2 𝑛 + 1 )
  • B 𝑓 ( 𝑛 ) = ( 2 𝑛 + 1 )
  • C 𝑓 ( 𝑛 ) = 2 ( 𝑛 1 )
  • D 𝑓 ( 𝑛 ) = 𝑛 ( 2 𝑛 1 ) = 2 𝑛 𝑛 2
  • E 𝑓 ( 𝑛 ) = 2 ( 𝑛 + 1 )

Q10:

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 2 3 4 ) .

  • A1, 1, 1
  • B0, 1, 1
  • C0, 1 , 1
  • D0, 0, 0
  • E1, 1 , 0

List the values of 𝑇 1 2 , 𝑇 3 2 , 𝑇 5 2 , and 𝑇 1 2 3 3 2 .

  • A1, 1 , 1, 1
  • B 1 , 1 , 1, 1
  • C1, 1, 1, 1
  • D1, 1 , 1, 1
  • E1, 1, 1 , 1

What is 𝑇 4 9 3 3 2 ?

  • A 1
  • B1
  • C0
  • Dundefined

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

  • A There is some integer 𝑛 for which 2 𝑛 + 1 < 𝑏 < 2 𝑛 + 2 .
  • B 𝑏 is an even integer.
  • C There is some integer 𝑛 for which 2 𝑛 < 𝑏 < 2 𝑛 + 1 .
  • D 𝑏 is an odd integer.

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

  • A 𝑦 = 1 2 ( 𝑥 1 2 )
  • B 𝑦 = 2 ( 3 𝑥 1 )
  • 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.

  • A16.850
  • B 0 . 5 6 6
  • C12.283
  • D 0 . 2 8 3
  • E4.429

Q11:

Given the sequence defined by 𝑇 = 𝑇 + 𝑛 𝑥 𝑛 + 1 𝑛 , where 𝑇 = 2 7 1 and 𝑇 = 7 8 3 , find the value of 𝑥 .

Q12:

Find the first five terms of the sequence with general term 𝑇 = 𝑇 1 3 𝑛 + 1 𝑛 , where 𝑛 1 and 𝑇 = 1 1 .

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

Q13:

Find the first five terms of the sequence with general term 𝑇 = 𝑇 3 𝑛 + 1 𝑛 , where 𝑛 1 and 𝑇 = 1 9 1 .

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

Q14:

Find the first five terms of the sequence with general term 𝑇 = 𝑇 + 2 6 𝑛 + 1 𝑛 , where 𝑛 1 and 𝑇 = 2 0 1 .

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

Q15:

Find the first five terms of the sequence with general term 𝑇 = 𝑇 + 3 𝑛 + 1 𝑛 , where 𝑛 1 and 𝑇 = 4 1 .

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

Q16:

The 𝑛 t h term in a sequence is given by 𝑇 = 𝑇 + 𝑇 𝑛 + 2 𝑛 + 1 𝑛 . Find the first six terms of this sequence, given that 𝑇 = 9 1 and 𝑇 = 1 1 1 2 .

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

Q17:

The 𝑛 t h term in a sequence is given by 𝑇 = 𝑇 + 𝑇 𝑛 + 2 𝑛 + 1 𝑛 . Find the first six terms of this sequence, given that 𝑇 = 1 6 3 1 and 𝑇 = 1 3 0 2 .

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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.