Question

Complete The Activity​

Answer 1

$\huge\mathfrak\green{ }$

given series is 0.7 + 0.77 + 0.777 + ......... to nth term

therefore ,

$S_n$= 7×(0.1 + 0.11 + 0.111 +......to nth term)

=7/9 ( 0.9 + 0.99 + 0.999 + ...... to nth term )

=7/9 ( 9/10 + 99/100 + 999/1000 + ..... to nth term )

= 7/9 { ( 1 - 1/10 ) + ( 1 - 1/100 ) + ( 1 - 1/1000 ) + .... to nth term }

= 7/9 { 1+1+1+....to nth term } - { 1/10 + 1/100 + 1/1000 +.... to nth term}

= 7/9 { n - 1/10 ( 1 - 1/10 ) ^n / 1 - 1/10 }

= 7/9 { n - 1 ( 1 - 1/10 )^n / 9 }

= 7/9 { n - ( 1 - 1/10 )^n / 9 }

$\huge\mathfrak\blue{ proved }$

Answer 2

Answer:

series is 0.7 + 0.77 + 0.777 + ____ to 9th term

= 7×(0.1 + 0.11 + 0.111 +___to 9th term)

=7/9 ( 0.9 + 0.99 + 0.999 +___to 9th term )

=7/9 ( 9/10 + 99/100 + 999/1000 + _____ to 9th term )

= 7/9 { ( 1 - 1/10 ) + ( 1 - 1/100 ) + ( 1 - 1/1000 ) + ____ to 9th term }

= 7/9 { 1+1+1+____to 9th term } - { 1/10 + 1/100 + 1/1000 +___ to 9th term}

= 7/9 { n - 1/10 ( 1 - 1/10 ) ^n / 1 - 1/10 }

= 7/9 { n - 1 ( 1 - 1/10 )^n / 9 }

= 7/9 { n - ( 1 - 1/10 )^n / 9 }

hence proved