Find k in JEE Mains Homework: Solving Equations with Geometric Progressions

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Homework Statement



Find k.

Homework Equations



If (10)9 + 2(11)1 (10)8 + 3(11)2 (10)7 + ...… + 10 (11)9 = k(10)9

The Attempt at a Solution



dividing both sides by 109,

1 + {2(11)1}/(10)1} + {3(11)2}/(10)2} + ..... + {10(11)9}/(10)9} = k

1(1.1)0 + 2(1.1)1 + 3(1.1)2 + ... + 10(1.1)9

∑(from n=0 to n=9) (n+1)(1.1)n
 
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Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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