# 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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Find a recursive formula for the sequence .

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Find a recursive formula for the sequence .

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Find a recursive formula for the sequence and .

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Q5:

Find the first five terms of the sequence with general term , where and .

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Q6:

The term in a sequence is given by . Find the first six terms of this sequence, given that and .

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Q7:

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

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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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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.

 𝑛 𝑓(𝑛) 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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Q24:

Find the first four terms in the sequence . Take , where ( an integer).

• A, , ,
• B, , ,
• C, , ,
• D, , ,
• E, , ,

Q25:

Write down a recursive formula for the sequence .

• A, ,
• B, ,
• C, ,
• D, ,
• E, ,