1,2,3....9 Wait, where is 10?

Algebra Level 5

If f ( x ) f(x) is a least degree polynomial such that f ( r ) = 1 r f(r)=\dfrac{1}{r} for r = 1 , 2 , 3 , … , 9 r=1,2,3,\ldots,9 , compute f ( 10 ) f(10) .

If your answer can be represented as m n \dfrac{m}{n} , where m m and n n are coprime positive integers, find m + n m+n .


The answer is 6.

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Shanthanu Rai by Shanthanu Rai , 21, India

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1 solution

Shanthanu Rai
Apr 1, 2016

r f ( r ) − 1 = 0 h a s r o o t s 1 , 2 , 3 , . . , 9 T h e r e f o r e , r f ( r ) − 1 = A ( r − 1 ) ( r − 2 ) . . . . ( r − 9 ) ( w h e r e A i s a c o n s t a n t ) P u t t i n g r = 0 ⟹ A = 1 9 ! F i n a l l y , p u t t i n g r = 10 ⟹ 10 f ( 10 ) − 1 = A × 9 ! ⟹ f ( 10 ) = 1 5 rf(r)-1=0 \: has \: roots \: 1,2,3,..,9 \\ Therefore, \: rf(r)-1=A(r-1)(r-2)....(r-9) \:\:\:\: (where \: A \: is \: a \: constant) \\ Putting \:r=0 \: \implies \:\:A=\frac{1}{9!} \\ Finally, \:putting \: r=10 \:\implies \:\:10f(10)-1=A\times 9! \\ \implies \:f(10)=\boxed{\frac{1}{5}}

12 Helpful 0 Interesting 0 Brilliant 0 Confused

Same approach! (+1)

Manuel Kahayon - 5 years, 2 months ago

Exactly the same solution!

A Former Brilliant Member - 5 years, 2 months ago

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