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If alpha and beta are the roots of the equation ax2 + bx + c = 0 and Ā Sn = (alpha)n + (beta)n, show that Ā aSn+1 + bSn Ā + cSn-1 = 0 and hence find S5.
alfa = p & beta = q aSn+1 +bSn + cSn-1 = 0 taking LHS = a(pn+1 + qn+1)+b(pn+qn)+c(pn-1+ qn-1) =[apn+1+bpn+cpn-1] + [aqn+1+bqn+cqn-1] =pn-1[ap2+bp+c] + qn-1[aq2+bq+c] now since p,q are the roots of eq ax2+bx+c=0 so p,q will satisfy this eq & ap2+bp+c = 0 = aq2+bq+c so the LHS becomes LHS = 0+0 =0 =RHS hence proved
alfa = p & beta = q
aSn+1 +bSn + cSn-1 = 0
taking LHS
= a(pn+1 + qn+1)+b(pn+qn)+c(pn-1+ qn-1)
=[apn+1+bpn+cpn-1] + [aqn+1+bqn+cqn-1]
=pn-1[ap2+bp+c] + qn-1[aq2+bq+c]
now since p,q are the roots of eq ax2+bx+c=0 so p,q will satisfy this eq &
ap2+bp+c = 0 = aq2+bq+c
so the LHS becomes
LHS = 0+0 =0 =RHS
hence proved
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