Question

$\blue{QUESTION}$
find sum to n term
6+66+666+...n
$\pink{ANSWER}$
$\frac{6}{81} ( {10}^{ n+ 1} - 9n - 10 )$

​

Answer 1

°°Question-

6+66+666+...n

••Answer

$Sn = 6 + 66 + 666 + ...n$

$= {6} (1 + 11 + 111 + ...n) \\ = \frac{6}{9}(9 + 99 + 999 + ...n) \\ = \frac{6}{9} {(10 - 1) + (100 - 1) + (1000 - 1) + ...n}$

$= \frac{6}{9} |(10 + 100 + 1000 + ...n) \\ - (1 + 1 + 1 + ...n)$

$= \frac{6}{9} (\frac{10( {10}^{ n- 1} )}{10 - 1} - n) \\ = \frac{6}{9} (\frac{ {10}^{ n+ 1} - 10 - 9n }{9} ) \\ = \frac{6}{81} ({10}^{ n+ 1} - 9n - 10).$

hope it's helpful

Answer 2

°°Question-

6+66+666+...n

••Answer

Sn = 6 + 66 + 666 + ...nSn=6+66+666+...n

\begin{gathered} = {6} (1 + 11 + 111 + ...n) \\ = \frac{6}{9}(9 + 99 + 999 + ...n) \\ = \frac{6}{9} {(10 - 1) + (100 - 1) + (1000 - 1) + ...n}\end{gathered}

=6(1+11+111+...n)

=

9

6

(9+99+999+...n)

=

9

6

(10−1)+(100−1)+(1000−1)+...n

\begin{gathered} = \frac{6}{9} |(10 + 100 + 1000 + ...n) \\ - (1 + 1 + 1 + ...n) \end{gathered}

=

9

6

∣(10+100+1000+...n)

−(1+1+1+...n)

\begin{gathered} = \frac{6}{9} (\frac{10( {10}^{ n- 1} )}{10 - 1} - n) \\ = \frac{6}{9} (\frac{ {10}^{ n+ 1} - 10 - 9n }{9} ) \\ = \frac{6}{81} ({10}^{ n+ 1} - 9n - 10).\end{gathered}

=

9

6

(

10−1

10(10

n−1

)

−n)

=

9

6

(

9

10

n+1

−10−9n

)

=

81

6

(10

n+1

−9n−10).

hope it's helpful