GATE Mechanical Engineering (ME) 2020 Shift 1 Solved Paper

Question : 9 of 65

Marks: +1, -0

The sum of the first

n

terms in the sequence

8,88,888,8888, . . .

Solution:

Sum of first

n

term is

=8+88+888+8888+. . . . .

=8[1+11+111+1111+. . . . .]

=\;{8}/{9}[9+99+999+9999+. . . . . .]

\;\; =\;{8}/{9}[(10-1)+(10^{2}-1)+(10^{3}-1)+(10^{4}-1)]

\;\; =\;{8}/{9}[10+10^{2}+10^{3}+10^{4} . . . . . . n]

\;\; =\;{8}/{9}[10+10^{2}+10^{3}+10^{4} . . . . n-(1+1 . . . . n)]

\;{8}/{9}[10 ⋅ \;{(10^{n}-1)}/{10-1}-n]

\;\; =\;{80}/{81}(10^{n}-1)-\;{8}/{9} n

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