Question

Simplify: 8 + 88 + 888 + 8888 + 88888​

Answer 1

Answer:

98760

Step-by-step explanation:

Consider the following composite series having  n  number of terms,

8+88+888+8888+…+(8888…n 8's)  

=8⎛⎝1+11+111+1111+…+(1111…n 1's)⎞⎠  

=89⎛⎝9+99+999+9999+⋯+(9999…n 9's)⎞⎠  

=89((10−1)+(102−1)+(103−1)+(104−1)+⋯+(10n−1))  

=89⎛⎝⎜⎜(10+102+103+…+10n)G.P. with n number of terms, first term 10 & common ratio 10−(1+1+1+1+…+1n 1's)⎞⎠⎟⎟  

=89(10(10n−1)10−1−n)  

=881(10(10n−1)−9n)  

Hence, the generalized formula to get sum of n terms of this composite series.

Now, substituting  n=5  in the above generalized formula for given series  8+88+888+8888+88888  having five terms, the sum is given as

8+88+888+8888+88888=881(10(105−1)−9⋅5)=98760  

Alternatively, one can fairly easily add all five terms of given series as follows

8+88+888+8888+88888––––––––=98760  

NOTE: The most generalized formula to get sum of n terms of such composite series is given as

 

Where,  x  is any digit i.e.  x∈{0,1,2,3,4,5,6,7,8,9}

Answer 2

Step-by-step explanation:

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