IXL - Convert between explicit and recursive formulas for geometric sequences (Algebra 2 practice)

Here is the initial term and recursive formula for a sequence:

(

)

300

(

)

(

–

)

Which explicit formula can be used to define this sequence?

(

)

300

–

(

)

300

297

–

(

)

300

303

(

)

300

ref_doc_title.

Learn with an example

question

Here is the initial term and recursive formula for a sequence:

(

)

(

)

–

(

–

)

Which explicit formula can be used to define this sequence?

(

)

–

(

)

(

)

(

)

(

–

)

–

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 variable

in 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.

f
(
1
)
=
40
f
(
2
)
=
–
f
(
1
)
=
–
1
40
f
(
3
)
=
–
f
(
2
)
=
–
1
–
1
40
f
(
4
)
=
–
f
(
3
)
=
–
1
–
1
–
1
40

Each term is the product of factors, one of which is 40, and the rest of which are

–

Rewrite the terms using exponents. (Recall that

and

(

–

)

f
(
1
)
=
40
=
(
–
1
)
0
40
f
(
2
)
=
–
1
40
=
(
–
1
)
1
40
f
(
3
)
=
–
1
–
1
40
=
(
–
1
)
2
40
f
(
4
)
=
–
1
–
1
–
1
40
=
(
–
1
)
3
40

Relate the expressions to the indices.

f
(
1
)
=
(
–
1
)
0
40
=
(
–
1
)
1
–
1
40
f
(
2
)
=
(
–
1
)
1
40
=
(
–
1
)
2
–
1
40
f
(
3
)
=
(
–
1
)
2
40
=
(
–
1
)
3
–
1
40
f
(
4
)
=
(
–
1
)
3
40
=
(
–
1
)
4
–
1
40

These terms satisfy the explicit formula

(

)

(

–

)

–

which is more commonly written as

(

)

(

–

)

–

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: Geometric sequences

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MM.2Find terms of a geometric sequence

Lesson: Geometric sequences

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