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求数5，55，555，…，55…5的前n项和Sn．
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求数5，55，555，…，55…5 的前n项和Sn．n
▼优质解答
答案和解析
∵55…5=
(10n −1)
∴Sn=5+55+555+…+55…5
=
[(10−1)+(102−1)+…+(10n−1)]
=
[
−n]
=
×10n−
n−
5 5 59 9 9(10n −1)
∴Sn=5+55+555+…+55…5
=
[(10−1)+(102−1)+…+(10n−1)]
=
[
−n]
=
×10n−
n−
n −1)
∴Snn=5+55+555+…+55…5
=
[(10−1)+(102−1)+…+(10n−1)]
=
[
−n]
=
×10n−
n−
5 5 59 9 9[(10−1)+(102−1)+…+(10n−1)]
=
[
−n]
=
×10n−
n−
2−1)+…+(10n−1)]
=
[
−n]
=
×10n−
n−
n−1)]
=
[
−n]
=
×10n−
n−
5 5 59 9 9[
10(10n−1) 10(10n−1) 10(10n−1)n−1)10−1 10−1 10−1−n]
=
×10n−
n−
50 50 5081 81 81×10n−
n−
n−
5 5 59 9 9n−
50 50 5081 81 81
| 5 |
| 9 |
∴Sn=5+55+555+…+55…5
=
| 5 |
| 9 |
=
| 5 |
| 9 |
| 10(10n−1) |
| 10−1 |
=
| 50 |
| 81 |
| 5 |
| 9 |
| 50 |
| 81 |
| 5 |
| 9 |
∴Sn=5+55+555+…+55…5
=
| 5 |
| 9 |
=
| 5 |
| 9 |
| 10(10n−1) |
| 10−1 |
=
| 50 |
| 81 |
| 5 |
| 9 |
| 50 |
| 81 |
∴Snn=5+55+555+…+55…5
=
| 5 |
| 9 |
=
| 5 |
| 9 |
| 10(10n−1) |
| 10−1 |
=
| 50 |
| 81 |
| 5 |
| 9 |
| 50 |
| 81 |
| 5 |
| 9 |
=
| 5 |
| 9 |
| 10(10n−1) |
| 10−1 |
=
| 50 |
| 81 |
| 5 |
| 9 |
| 50 |
| 81 |
=
| 5 |
| 9 |
| 10(10n−1) |
| 10−1 |
=
| 50 |
| 81 |
| 5 |
| 9 |
| 50 |
| 81 |
=
| 5 |
| 9 |
| 10(10n−1) |
| 10−1 |
=
| 50 |
| 81 |
| 5 |
| 9 |
| 50 |
| 81 |
| 5 |
| 9 |
| 10(10n−1) |
| 10−1 |
=
| 50 |
| 81 |
| 5 |
| 9 |
| 50 |
| 81 |
| 50 |
| 81 |
| 5 |
| 9 |
| 50 |
| 81 |
| 5 |
| 9 |
| 50 |
| 81 |
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