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a(n+1)=an+2n+1
=[a(n-1)+2(n-1)+1]+(2n+1)
=a(n-1)+[2(n-1)+1]+(2n+1)
=[a(n-2)+2(n-2)+1]+[2(n-1)+1]+(2n+1)
=a(n-2)+[2(n-2)+1]+[2(n-1)+1]+(2n+1)
=a1+(2*1+1)+(2*2+1)+...+(2n+1)
=1+2*(1+n)n/2+n=1+n+n**2+n
=1+2n+n**2
∴当n>1时,
an=1+2(n-1)+(n-1)**2=(n-1)**2+2n-1
∵a1=1
∴n=1时,an=1
n>1时,an=(n-1)**2+2n-1
=[a(n-1)+2(n-1)+1]+(2n+1)
=a(n-1)+[2(n-1)+1]+(2n+1)
=[a(n-2)+2(n-2)+1]+[2(n-1)+1]+(2n+1)
=a(n-2)+[2(n-2)+1]+[2(n-1)+1]+(2n+1)
=a1+(2*1+1)+(2*2+1)+...+(2n+1)
=1+2*(1+n)n/2+n=1+n+n**2+n
=1+2n+n**2
∴当n>1时,
an=1+2(n-1)+(n-1)**2=(n-1)**2+2n-1
∵a1=1
∴n=1时,an=1
n>1时,an=(n-1)**2+2n-1
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