IXL - Convert recursive formulas to explicit formulas for arithmetic sequences: function notation (Algebra 1 practice)

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

(

)

–

(

)

(

–

)

–

Which explicit formula can be used to define this sequence?

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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
)
=
28
f
(
2
)
=
f
(
1
)
–
5
=
28
–
5
f
(
3
)
=
f
(
2
)
–
5
=
28
–
5
–
5
f
(
4
)
=
f
(
3
)
–
5
=
28
–
5
–
5
–
5

Each term is the sum of numbers, one of which is 28, and the rest of which are

–

Rewrite the terms using multiplication. (Recall that

–

f
(
1
)
=
28
=
28
–
5
0
f
(
2
)
=
28
–
5
=
28
–
5
1
f
(
3
)
=
28
–
5
–
5
=
28
–
5
2
f
(
4
)
=
28
–
5
–
5
–
5
=
28
–
5
3

Relate the expressions to the indices.

f
(
1
)
=
28
–
5
0
=
28
–
5
(
1
–
1
)
f
(
2
)
=
28
–
5
1
=
28
–
5
(
2
–
1
)
f
(
3
)
=
28
–
5
2
=
28
–
5
(
3
–
1
)
f
(
4
)
=
28
–
5
3
=
28
–
5
(
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

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U.2Arithmetic sequences

Lesson: Arithmetic sequences

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