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某下崗職工準(zhǔn)備開(kāi)辦一個(gè)商店，要向銀行貸款若干，這筆貸款按復(fù)利計(jì)算(即本年利息計(jì)入下一年的本金生息)，利率為q(0＜q＜1)．據(jù)他估算，貸款后每年可償還A元，30年后還清．

①求貸款金額；

②若貸款后前7年暫不償還，從第8年開(kāi)始，每年償還A元，仍然在貸款30年還清；試問(wèn)：這樣一來(lái)，貸款金額比原貸款金額要少多少元？

答案：
解析：

思路　　團(tuán)貸款按復(fù)利計(jì)算，所以貸款后每?jī)斈瓯�、息不相�?/TITLE> </HEAD> <BODY TEXT="#000000"> <P ALIGN="JUSTIFY">　　思路　　團(tuán)貸款按復(fù)利計(jì)算，所以貸款后每?jī)斈瓯尽⑾⒉幌嗤撀毠で房钪鹉隃p少，直到30年后還清，對(duì)此問(wèn)題可建立數(shù)列模型．</P> <P ALIGN="JUSTIFY">　　解答　�、僭O(shè)貸款金額為x元，則</P> <P ALIGN="JUSTIFY">　　1年后欠款為a<SUB>1</SUB>＝x(1＋q)－A(元)</P> <P ALIGN="JUSTIFY">　　2年后欠款為a<SUB>2</SUB>＝a<SUB>1</SUB>(1＋q)－A＝x(1＋q)<SUP>2</SUP>－A[(1＋q)＋1]，</P> <P ALIGN="JUSTIFY">　　3年后欠款為a<SUB>3</SUB>＝a<SUB>2</SUB>(1＋q)－A＝x(1＋q)<SUP>3</SUP>－A[(1＋q)<SUP>2</SUP>＋(1＋q)＋1]，</P> <P ALIGN="JUSTIFY">　　仿此，30年后欠款為</P> <P ALIGN="JUSTIFY">　　a<SUB>30</SUB>＝x(1＋q)<SUP>30</SUP>－A[(1＋q)<SUP>29</SUP>＋(1＋q)<SUP>28</SUP>＋…＋1]．</P> <P ALIGN="JUSTIFY">　　因?yàn)閍<SUB>30</SUB>＝0，所以x(1＋q)<SUP>30</SUP>＝A<IMG align="middle" SRC="http://thumb.1010pic.com/pic7/pages/60A1/0005/0471/b050e2bf9fe065178aeb300cefbfbb04/C/Image35459.gif" width=93 height=50>，</P> <P ALIGN="JUSTIFY">　　所以x＝<IMG style="vertical-align:middle" SRC="http://thumb.1010pic.com/pic7/pages/60A1/0005/0471/b050e2bf9fe065178aeb300cefbfbb04/C/Image35460.gif" width=20 HEIGHT=44>·<IMG style="vertical-align:middle" SRC="http://thumb.1010pic.com/pic7/pages/60A1/0005/0471/b050e2bf9fe065178aeb300cefbfbb04/C/Image35461.gif" width=80 height=48>(元)；</P> <P ALIGN="JUSTIFY">　�、谠O(shè)第8年開(kāi)始償還的這種貸款金額為y元，則</P> <P ALIGN="JUSTIFY">　　8年后欠款為b<SUB>8</SUB>＝y(tǒng)(1＋q)<SUP>8</SUP>－A，</P> <P ALIGN="JUSTIFY">　　9年代欠款為b<SUB>9</SUB>＝b<SUB>8</SUB>(1＋q)－A＝y(tǒng)(1＋q)<SUP>9</SUP>－A[(1＋q)＋A]．</P> <P ALIGN="JUSTIFY">　　仿此，30年后欠款為</P> <P ALIGN="JUSTIFY">　　b<SUB>30</SUB>＝y(tǒng)(1＋q)<SUP>30</SUP>－A[(1＋q)<SUP>22</SUP>＋(1＋q)21＋…＋1]</P> <P ALIGN="JUSTIFY">　　因?yàn)閎<SUB>30</SUB>＝0，所以y(1＋q)<SUP>30</SUP>＝A[(1＋q)<SUP>22</SUP>＋(1＋q)<SUP>21</SUP>＋…＋1]，</P> <P ALIGN="JUSTIFY">　　所以y＝<IMG style="vertical-align:middle" SRC="http://thumb.1010pic.com/pic7/pages/60A1/0005/0471/b050e2bf9fe065178aeb300cefbfbb04/C/Image35462.gif" width=20 HEIGHT=44>·<IMG style="vertical-align:middle" SRC="http://thumb.1010pic.com/pic7/pages/60A1/0005/0471/b050e2bf9fe065178aeb300cefbfbb04/C/Image35463.gif" width=80 height=48>(元)</P> <P ALIGN="JUSTIFY">　　故x－y＝<IMG style="vertical-align:middle" SRC="http://thumb.1010pic.com/pic7/pages/60A1/0005/0471/b050e2bf9fe065178aeb300cefbfbb04/C/Image35464.gif" width=20 HEIGHT=44>·<IMG style="vertical-align:middle" SRC="http://thumb.1010pic.com/pic7/pages/60A1/0005/0471/b050e2bf9fe065178aeb300cefbfbb04/C/Image35465.gif" width=89 height=50>(元)．</P> <P ALIGN="JUSTIFY">　　評(píng)析　　從數(shù)學(xué)角度看，本例是解決與數(shù)列有關(guān)的應(yīng)用問(wèn)題．必須認(rèn)真審題，弄清題意，解決問(wèn)題的關(guān)鍵在于理解復(fù)利的概念及其運(yùn)算，形成用數(shù)學(xué)的意識(shí)．</P> </BODY> </HTML> </td></tr></table><br> </div> </div> <div id="geuiha1" class="sublist"><div id="wr0unig" class="answer_inner pic-group" id="daan_recommend"><img src="http://thumb2018.1010pic.com/images/loading.gif" /></div></div> <div id="esmlqxz" class="sublist"><div id="lead8uv" class="answer_inner" 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