# 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 .

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**Q2: **

Write a recursive formula for the following sequence that is valid for :

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**Q3: **

Using the absolute value function, write a recursive formula for the following sequence that is valid for :

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**Q4: **

Find a recursive formula for the sequence .

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- B ,
- C ,
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- E ,

Find a recursive formula for the sequence .

- A ,
- B ,
- C ,
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- E ,

Find a recursive formula for the sequence .

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- B ,
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Find a recursive formula for the sequence and .

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**Q8: **

Given the sequence defined by , where and , find the value of .

**Q9: **

Find given and .

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**Q10: **

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

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**Q11: **

Find the arithmetic sequence in which and .

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**Q12: **

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

List the values of , , and .

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

List the values of , , , and .

- A1, , 1,
- B1, 1, 1, 1
- C , , 1, 1
- D1, 1, , 1
- E1, , 1, 1

What is ?

If we are given that is negative, what can we conclude about the number ?

- AThere is some integer for which .
- B is an even integer.
- CThere is some integer for which .
- D is an odd integer.

Find the equation of the line segment on which the point lies.

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Hence find the value of correct to 3 decimal places.

**Q13: **

Find, in terms of the general term of the sequence which satisfies the relation , where and .

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**Q14: **

Consider the sequence given by .

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?

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**Q15: **

Find the relationship between the terms in the sequence .

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**Q16: **

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

Let be the th positive solution to the equation . Starting from , write a recursive formula for .

- A for .
- B for .
- C for .
- D for .
- E for .

What is the set of numbers which satisfy the equation

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The part of the graph through the origin coincides with the line . Use this to find one solution to . Use the symmetries of the graph to find the next positive solution.

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- B .
- C .
- D .
- E .

Find the first two positive solutions to .

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Find the value of , giving your answer correct to 3 decimal places.

**Q17: **

The sequence is given by a recursive formula of the form for . Determine this formula.

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**Q18: **

Find the first six terms of the sequence satisfying the relation , where , , and .

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**Q19: **

The function is defined by and for . Complete the table.

1 | 2 | 3 | 4 | |

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**Q20: **

Find given , , and .

**Q21: **

Find the first five terms of the sequence , given , , and .

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**Q22: **

Consider the sequence .

Find a recursive formula for this sequence.

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The recursive formula , where , can be made explicit with a formula of the form for suitable numbers and . Find the values of and .

- A and
- B and
- C and
- D and
- E and

**Q23: **

A sequence is defined by the recursive formula .

Find the first six terms of this sequence.

- A , , , , ,
- B , , , , ,
- C , , , , ,
- D , , , , ,
- E , , , , ,

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

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

Find an explicit formula for , where .

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

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Consider the sequence defined by the recursive formula . Find the value of for which is a geometric sequence with common ratio 7.

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

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