Question

Find an explicit formula for the sequence 2, 4, 8, 16, 32

Answer 1

Answer:

a

n

=

4

⋅

2

n

−

1

if it is a geometric sequence.

or could be:  

a

n

=

1

3

(

2

n

3

−

6

n

2

+

16

n

)

if not.

Explanation:

There is a common ratio between successive pairs of terms:

8

4

=

2

16

8

=

2

32

16

=

2

So this looks like a geometric sequence with initial term  

a

=

4

and common ratio  

r

=

2

.

If so, the formula for the  

n

th term is:

a

n

=

a

r

n

−

1

=

4

⋅

2

n

−

1

This is probably the answer expected by the questioner.

However, note that any finite sequence of terms does not determine an infinite sequence - unless you are told what kind of sequence it is - e.g. arithmetic, geometric, harmonic.

For example, we can match these first  

4

terms with a cubic formula as follows:

Write down the sequence as a list:

4

,

8

,

16

,

32

Write down the sequence of differences between each pair of terms:

4

,

8

,

16

Write down the sequence of differences of this sequence:

4

,

8

Write down the sequence of differences of this sequence:

4

Having reached a constant sequence (albeit consisting of only one term), we can use the initial term of each of the sequences we have found as coefficients of a formula for the  

n

th term:

a

n

=

4

0

!

+

4

1

!

(

n

−

1

)

+

4

2

!

(

n

−

1

)

(

n

−

2

)

+

4

3

!

(

n

−

1

)

(

n

−

2

)

(

n

−

3

)

=

4

+

(

4

n

−

4

)

+

(

2

n

2

−

6

n

+

4

)

+

(

2

3

n

3

−

4

n

2

+

22

3

n

−

4

)

=

2

3

n

3

−

2

n

2

+

16

3

n

=

1

3

(

2

n

3

−

6

n

2

+

16

n

Step-by-step explanation: