Recursive Formula

[albswb]

I am stuck with this problem, please help!

Write a recursive formula for the sequence {1, 3, 7, 15, 31, ...}

This is what I have done:

1x2=2+1=3
3x2=6+1=7
7x2=14+1=15
etc.

a1=1
an=2a(n-1) +1 for all n>2

Is this correct?

Thanks so much!

 

[soroban]

Hello, albswb!

Write a recursive formula for the sequence {1, 3, 7, 15, 31, ...}

This is what I have done:

. . 1 x 2 + 1 = 3
. . 3 x 2+ 1 = 7
. . 7 x 2 + 1 = 15
. . . . etc.

\(\displaystyle a_1\,=\,1\)
\(\displaystyle a_n\:=\:2a_{n-1}+1\;\text{ for all }n>2\)

Is this correct? . Yes!

The closed form is: .\(\displaystyle a_n \:=\:2^n - 1\)

 

[HallsofIvy]

I am stuck with this problem, please help!

Write a recursive formula for the sequence {1, 3, 7, 15, 31, ...}

This is what I have done:

1x2=2+1=3

I don't much like your notation! 1x2 is NOT equal to 3. rather write 1x2+1= 2+1= 3

3x2=6+1=7

You mean 3x2+1= 6+1= 7

7x2=14+1=15

You mean 7x2+1= 15

etc.

a1=1
an=2a(n-1) +1 for all n>2

Is this correct?

Thanks so much!

Yes, that is correct.