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

Question

If ${(10)}^{9} + 2 {(11)}^{1} {(10)}^{8} + 3 {(11)}^{2} {(10)}^{7} + \dots + 10 {(11)}^{9} = k {(10)}^{9}$ , then $k$ is equal to

Smart Achievers · Accepted Answer

(2)

$Let P = 10^{9} + 2 \cdot 11 \cdot 10^{8} + \dots + 10 \cdot 11^{9}$

$We have, \frac{11}{10} P = 11 \cdot 10^{8} + \dots + 9 \cdot 11^{9} + 11^{10}$

$k$

$\Rightarrow \frac{1}{10} P = 11^{10} - [10^{9} + 11^{1} \cdot 10^{8} + 11^{2} \cdot 10^{7} + \dots + 11^{9}]$

$= 11^{10} - 10^{9} {\frac{{(\frac{11}{10})}^{10} - 1}{\frac{11}{10} - 1}} = 11^{10} - 11^{10} + 10^{10}$

$∴$ $P = 10^{11} = (100) \cdot 10^{9}$

$On comparison, k = 100$

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