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Mathematics
Problem
Consider a geometric sequence with first term 1 and common ratio 10.
S
n
S_n
S
n
is the sum of the first
n
n
n
terms of the sequence. a. Find an expression for
S
n
S_n
S
n
in the form
a
n
−
1
b
\frac{a^n-1}{b}
b
a
n
−
1
, where
a
,
b
∈
Z
+
a,b\in\mathbb{Z}^+
a
,
b
∈
Z
+
. b. Hence, show that
S
1
+
S
2
+
S
3
+
⋯
+
S
n
=
10
(
10
n
−
1
)
−
9
n
81
.
S_1+S_2+S_3+\dots+S_n=\frac{10(10^n-1)-9n}{81}.
S
1
+
S
2
+
S
3
+
⋯
+
S
n
=
81
10
(
1
0
n
−
1
)
−
9
n
.
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