compare the ratio 3 : 33 and 33 : 333
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From the given series, take common factor 3
Thus
S=3(1+11+111+1111+...............nthterm)
Or
S=3{1+(1+10)+(1+10+100)+(1+10+100+1000)+...tonterms}
Thus the nth term.
t
n
=1+10+100+..............+10
n−1
This is a G.P. where first term is 1 and common ratio is 10 ,
Thus sum of n terms is
t
n
=
1−10
1(1−10
n
)
t
n
=
9
(10
n
−1)
Or
Thus
S=3∑
i=1
n
t
n
=3∑
i=1
n
9
10
n
−1
Hence the sum,
S=3∑
9
10
n
−3∑
9
1
Or
S=3{
(1−10).9
10(1−10
n
)
−(
9
1
).n}
Or
S=3{10.
81
(10
n
−1)
−
9
n
}
Thus,
S={10.
27
(10
n
−1)
−
3
n
}
=
27
10
n+1
−10−9n
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