IXL - Convert a recursive formula to an explicit formula (Precalculus practice)

Use the initial term and the recursive formula to find an explicit formula for the sequence

Write your answer in simplest form.

–

ref_doc_title.

Learn with an example

question

Use the initial term and the recursive formula to find an explicit formula for the sequence

Write your answer in simplest form.

–

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

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.

a
1
=
–
9
a
2
=
a
1
+
4
=
–
9
+
4
a
3
=
a
2
+
4
=
–
9
+
4
+
4
a
4
=
a
3
+
4
=
–
9
+
4
+
4
+
4

Each term is the sum of numbers, one of which is

–

and the rest of which are 4. Rewrite the terms using multiplication. (Recall that

–

a
1
=
–
9
=
–
9
+
4
0
a
2
=
–
9
+
4
=
–
9
+
4
1
a
3
=
–
9
+
4
+
4
=
–
9
+
4
2
a
4
=
–
9
+
4
+
4
+
4
=
–
9
+
4
3

Relate the expressions to the indices.

a
1
=
–
9
+
4
0
=
–
9
+
4
(
1
–
1
)
a
2
=
–
9
+
4
1
=
–
9
+
4
(
2
–
1
)
a
3
=
–
9
+
4
2
=
–
9
+
4
(
3
–
1
)
a
4
=
–
9
+
4
3
=
–
9
+
4
(
4
–
1
)

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:

–

Looking for more? Try these:

Lesson: Arithmetic sequences

Lesson: Geometric sequences

$Confetti$

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Teacher tools

Work it out

Not feeling ready yet? These can help:

AA.5Find recursive and explicit formulas

AA.2Find terms of a recursive sequence

Lesson: Arithmetic sequences

Lesson: Geometric sequences

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