Question

Find the sum to n terms of the sequence, 8, 88,888,8888.....
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Answer 1

GP : 8 , 88 , 888 , .......

= 8(1) + 8(11) + 8(111) ,......

Taking 8 common ,

= 8 ( 1 + 11 + 111 + ...... )

Divide and multiply by 9 ;

= 8/9 [ 9 + 99 + 999 + ..... ]

= 8/9 [ (10 - 1) + (100 - 1) + (1000 - 1) + ..... ]

= 8/9 [ (10 - 1) + (10² - 1) + (10³ - 1) + ..... ]

= 8/9 [ (10 + 10² + 10³ + .....) - ( 1 + 1 + 1 + .....) ]    { Applying GP formula }

= 8/9 [ ( 10 ( 10n - 1) / 10 - 1  )  -  1 ]

= 8/9 [ ( 10 ( 10n - 1 ) / 9 ) - n]

Sum of GP formula :

$\frac{a(r^n-1)}{r-1}$