# derivation of recurrence for Sylvester’s sequence

Let us begin with the product:

 $a_{n}=1+\prod_{i=0}^{n-1}a_{i}$

Adding $1$ to $n$ and manipulating the result:

 $\displaystyle a_{n+1}$ $\displaystyle=$ $\displaystyle 1+\prod_{i=0}^{n}a_{i}$ $\displaystyle=$ $\displaystyle 1+a_{n}\prod_{i=0}^{n-1}a_{i}$ $\displaystyle=$ $\displaystyle 1+a_{n}(a_{n}-1)=1+(a_{n})^{2}-a_{n}$
Title derivation of recurrence for Sylvester’s sequence DerivationOfRecurrenceForSylvestersSequence 2013-03-22 15:48:27 2013-03-22 15:48:27 rspuzio (6075) rspuzio (6075) 4 rspuzio (6075) Derivation msc 11A55