If α,β are roots of $$375x2$$−$$25x$$−$$2=0$$ and $$Sn=$$α$$n+$$βn , then limn→ ∞ ∑$$r=1nSr$$ is

Question

If  α,β      are roots of $$375x2$$−$$25x$$−$$2=0$$      and $$Sn=$$α$$n+$$βn     , then limn→ ∞ ∑$$r=1nSr$$      is

Infinity Learn · Accepted Answer

The given equation is $$375x2$$−$$25x$$−$$2=0$$ α,β are the roots of the equation, thenSum of the roots : α$$+$$β$$=25375=115$$  andProduct of the roots : αβ$$=$$−$$2375Given$$ that, $$Sn=$$α$$n+$$βn                     ∑$$r=1nSr=($$α$$+$$α$$2+$$.....$$+$$α$$n)+($$β$$+$$β$$2+$$....$$+$$β$$n)$$∴  limn→∞ ∑$$r=1nSr=($$α$$+$$α$$2+$$..... to  ∞$$)+($$β$$+$$β$$2+$$..... to  ∞$$)$$                     $$=$$α$$1$$−α$$+$$β$$1$$−β                 $$($$∵α$$+$$α$$2+$$..... to  ∞$$=$$α$$1$$−α   β$$+$$β$$2+$$....... to  ∞$$=$$β$$1$$−β$$)$$                       $$=$$α−αβ$$+$$β−αβ(1$$−α$$)(1$$−β$$)=($$α$$+$$β$$)−$$2$$αβ$$1$$−α−β$$+$$αβ                     $$=115+43751$$−$$115$$−$$2375=25+4375$$−$$25$$−$$2=29348=112$$

If α,β are roots of 375x2−25x−2=0 and Sn=αn+βn , then limn→ ∞ ∑r=1nSr is

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