a1 = 1 × a1
a2 = 4 × a1
a3 = 10 × a1
a4 = 20 × a1
a5 = 35 × a1
a2-a1=3 a1
a3-a2=6 a1
a4-a3=10 a1
a5-a4=15 a1
take bn=an+1-an
b1=3a1= a1 + 2a1
b2=b1 + 3a1
b3=b2 + 4a1
b4=b3 + 5a1
(plus all)
(a1=6)
→bn=(1+2+3+...+(n+1))a1=(1+(n+1))(n+1)/2 × 6=3(n+2)(n+1)=an+1-an
Then
a2-a1 = 3 × (3 × 2)
a3-a2 = 3 × (4 × 3)
a4-a3 = 3 × (5 × 4)
(plus all)
an-a1= 3 × sigma(n=1~n-1) { (n+2)(n+1) }= 3 × sigma(n=1~n-1) { n²+3n+2 }
an=3× { (n-1)n(2n-1)/6+ 3 × (1+n-1)(n-1)/2+2(n-1) } +6
=1/2 {2n³-3n²+n+9n²-9n+12n-12} +6
=1/2 {2n³+6n²+4n-12} +6
=n³+3n²+2n
