If (10)9 + 2(11)1(10)8 + 3(11)2(10)7 + ............. + 10(11)9 = k (10)9, then k is equal to

Engineering

Mathematics

Arithmetic Geometric Progression

Question

If (10)⁹ + 2(11)¹(10)⁸ + 3(11)²(10)7 + ............. + 10(11)⁹ = k (10)⁹, then k is equal to

$\frac{441}{100}$

100

$\frac{121}{10}$

110

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$10^{9} (1 + 2. (\frac{11}{10}) + 3. {(\frac{11}{10})}^{2} + .... + 10. {(\frac{11}{10})}^{9})$

Let k = 1 + 2x + 3x² + .... + 10x⁹

where $x = \frac{11}{10}$

$k = \frac{1 - x^{10}}{{(1 - x)}^{2}} - 10 . \frac{x^{10}}{1 - x}$

k = 100