IXL - Find recursive and explicit formulas (Precalculus practice)

These are the first six terms of a sequence starting with

–

103

–

118

–

133

...

Find a recursive and an explicit formula for this sequence.

The recursive formula should depend on

–

Recursive formula:

Explicit formula:

ref_doc_title.

Learn with an example

question

These are the first six terms of a sequence starting with

...

Find a recursive and an explicit formula for this sequence.

The recursive formula should depend on

–

Recursive formula:

Explicit formula:

key idea

A sequence is an ordered list of numbers, called

terms.

The terms in a sequence are sometimes written

etc., where

is the first term,

is the second term, and so on. The subscript

on the term

is called the

index.

A recursive formula defines a sequence using earlier terms.

An explicit formula defines a sequence using only the index,

solution

Find the recursive formula.

Start by looking at the differences of consecutive terms. In this case, the differences between consecutive terms are all the same. (Notice that

–

has no value for

since there is no

The first term in the sequence is

n	a n	a n – a n – 1
2	34	5
3	39	5
4	44	5
5	49	5
6	54	5

The table shows that

–

for

≥

So, the recursive formula is:

–

Find the explicit formula.

It is helpful to use the recursive formula to find an explicit formula. Use the initial term

and the recursive formula to write the first few terms of this sequence. Sometimes, the pattern can be easier to see if you don't simplify the terms.

Each term is the sum of numbers, one of which is 29, and the rest of which are 5. Rewrite the terms using multiplication. (Recall that

a
1
=
29
=
29
+
5
0
a
2
=
29
+
5
=
29
+
5
1
a
3
=
29
+
5
+
5
=
29
+
5
2
a
4
=
29
+
5
+
5
+
5
=
29
+
5
3

Relate the expressions to the indices.

(

–

)

(

–

)

(

–

)

(

–

)

The first few terms satisfy the explicit formula

(

–

)

Use substitution to verify that this explicit formula satisfies the recursive formula.

≟

–

≟

[

(

–

)

]

≟

–

This verifies that

satisfies the recursive formula

–

The explicit formula is:

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Work it out

Not feeling ready yet? These can help:

AA.4Find a recursive formula

MM.6Write a formula for an arithmetic sequence

MM.7Write a formula for a geometric sequence

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