Question

Pn+2=15Pn-1+3(Pn+Pn+1)

Answer 1

(n+2)!=15(n-1)!+3n! +3* (n+1)!
simplificam cu (n-1)!
 n(n+1)(n+2)=15+3n+3n(n+1)
n(n+1) (n+2)=15+3n(n+2) | :[n(n+2)]
n=15/(n(n+2))+3
n-3=15/n(n+2)
 (n-3)n(n+2)=15=1*3*5

n=3 cu conventia din combinatorica  (n-3)!=0!=1

Deci n=3

 Verificare
n=3
n-1=2
n+2=5
n+1=4
 
 P5=15P2+3(P3+P4)
stim ca
 P2=2!=2
P3=3!=6
P4=4!=24
 P5=5!=120
120=15*2+3(6+24)
120=30+3*30
120=30+90
120=120, adevarat ,  problema este bine rezolvata