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Question

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11459136425(n21)2n2+(n+1)2=(n+1)(n+2)(2n+3)6.

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Solution

Here, (n21)2n2+(n+1)2is the (n+1)th component
un+1={n2+(n+1)2}un(n21)2un1
un+1n2un=(n+1)2{un(n1)2un1}
un(n1)2un=n2{un1(n2)2un2}
...............................................................................................
u34u2=a(u2u1)

Hence by multiplication we obtain
un+1n2un=32.42.....(n+1)2(u2u1)
Now, p1=1,q1=1;p2=5,q2=1;
qn+1n2qn=0
qn+1=n2qn
qn+1=n2(n1)2....12=(n!)2
pn+1n2pn=32.42.....(n+1)24
pn+1(n!)2pn((n1)!)2=(n+1)2;
p2(1!)2p21=22;
By addition;-
pn+1(n!)21=22+32+.....+(n+1)2
pn+1(n!)2=12+22+.....+(n+1)2
qn+1(n!)2=1
pn+1qn+1=n2=(n+1)(n+2)(2n+3)6

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