Why does this sequence happen like this?

Question

The other day I sent my girlfriend this text

<3

she sent me back

<3<3<3

not to be one upped I responded with

<3<3<3<3<3<3<3

this got very silly very quickly.

After our "<3" battle was over I got to thinking about the pattern we were forming. Since we were doubling the number of hearts and adding one I thought the sequence would be something like 2^n + 1. BUT IT IS NOT. It is 2^n - 1 .

Why is this?

Because if you have $2^k + d$, double it and add one, you get $2^{k+1} + 2d + 1$. The difference is maintained iff it is $d = -1$. — Daniel Fischer, Jul 03 '13 at 18:08

score 5 · Accepted Answer · answered Jul 03 '13 at 18:08

5

Because you started with $1=2^1-1$. Then $2*1+1=2*(2^1-1)+1=2^2-1$. Generally $2(2^n-1)+1=2^{n+1}-1$ When you double the $-1$ and add $1$, you get $-1$ back again.

answered Jul 03 '13 at 18:08

Ross Millikan

374,822

score 2 · Answer 2 · answered Jul 03 '13 at 18:09

2

The pattern clearly holds for the first few terms. Let's see if going from step $n$ to step $n+1$ keeps with the pattern.

$a_n=2^n-1$. Then $$ a_{n+1}=2a_n+1=2(2^n-1)+1=2^{n+1}-2+1=2^{n+1}-1. $$

answered Jul 03 '13 at 18:09

Ian Coley

6,000

Why does this sequence happen like this?

2 Answers2