等差数列（an）中，Sn=30,S2n=100,则S3n= 为什么Sn S2n-Sn S 在等差数列中,已知sn=30,s2n=100则s3n=?
\u7b49\u5dee\u6570\u5217{an}\u7684\u524dn\u9879\u548c\u4e3aSn\uff0c\u82e5Sn=30\uff0cS2n=100\uff0c\u5219S3n=\uff08\u3000\u3000\uff09A\uff0e130B\uff0e170C\uff0e210D\uff0e26\u56e0\u4e3a\u6570\u5217{an}\u4e3a\u7b49\u5dee\u6570\u5217\uff0c\u6240\u4ee5\u7531\u7b49\u5dee\u6570\u5217\u6027\u8d28\u53ef\u5f97\uff1asn\uff0cs2n-sn\uff0cs3n-s2n\u2026\u4e3a\u7b49\u5dee\u6570\u5217\uff0e\u537330\uff0c100-30\uff0cS3n-100\u662f\u7b49\u5dee\u6570\u5217\uff0c\u22342\u00d770=30+S3n-100\uff0c\u89e3\u5f97S3n=210\uff0c\u6545\u9009C\uff0e
\u7279\u6b8a\uff1a\u4ee4n=1,a1=30,a2=70,\u5219a3=110,s3n=210
\u4e00\u822c\uff1as2n-sn=70,\u697c\u4e0a\u8bf4\u7684\u5e94\u662f\u8fd9\u4e2a30,70,\u2026\u2026\u4e0b\u4e00\u4e2a\u662f110
s3n=s2n+(s3n-s2n)=100+110=210
S2n-Sn
S3n-S2n成等差数列吧
其实这个比较简单我们把公式列一下
Sn=a1n+n(n-1)d/2
S2n=2a1n+n(2n-1)d
S3n=3a1n+3n(3n-1)d/2
两两互减
（S2n-Sn）-Sn=2a1n+n(2n-1)d-2a1n+n(n-1)d=n^2d
（S3n-S2n）-(S2n-Sn)=S3n-2S2n+Sn=3a1n+3n(3n-1)d/2-4a1n-2n(2n-1)d+a1n+n(n-1)d/2
=n^2d
所以,Sn,S2n-Sn,S3n-S2n成等差数列
考试的时候我们可以不用像上面一样一步步证明，可以把结论直接拿出来！
绛旓細鍥犱负鏁板垪{an}涓绛夊樊鏁板垪,鎵浠ョ敱绛夊樊鏁板垪鎬ц川鍙緱:sn,s2n-sn,s3n-s2n鈥︿负绛夊樊鏁板垪.鍗30,100-30,S3n-100鏄瓑宸暟鍒,鈭2脳70=30+S3n-100,瑙ｅ緱S3n=210,鏁呴塁.
绛旓細a3=a1+(3-1)d=a1+2d=6 a6=a1+(6-1)d=a1+5d=0 瑙ｅ緱a1=10 d=-2 sn=n*a1+n(n-1)d/2=10n-n(n-1)=30 寰梟^2-11n+30=0 (n-5)(n-6)=0 n=5鎴杗=6
绛旓細鎵浠n,S(2n)-Sn,S(3n)-S(2n)绛夊樊 30锛60锛孲(3n)-90绛夊樊 鎵浠3n=180
绛旓細Sn=30锛汼2n-Sn=60锛汼3n-S2n=S3n-90锛涳紙1锛绛夊樊锛2脳60=30+锛圫3n-90锛塖3n=180 锛2锛夌瓑姣旓細60^2=30脳锛圫3n-90锛塖3n=210
绛旓細(1)an =Sn -S(n-1)=31-2n n=1鏃朵篃鎴愮珛銆傛槸绛夊樊鏁板垪銆傚叕宸负-2 棣栭」涓29 閫氶」鍏紡an =31-2n (2)Sn=-n^2 +30n =-(n-15)^2 +225 鎵浠ワ紝褰搉=15鏃, Sn鏈澶у间负225.
绛旓細b/a路a/b)=4 褰撲笖浠呭綋 a = b = 2 鏃跺彇绛夊彿 鍗 a = b = 2 鏃 a+b鐨勬渶灏忓兼槸4 (2) 鈭祘an}鏄瓑宸暟鍒 鈭 Sn 锛孲2n - Sn 锛孲3n - S3n 涔熸垚绛夊樊鏁板垪锛鍏樊鏄 n^2 d 锛夊嵆 30,70,S3n -100 鎴愮瓑宸暟鍒 鈭 S3n -100 = 110 瑙ｅ緱 S3n = 210 ...
绛旓細杩欐槸涓掑噺鐨绛夊樊鏁板垪锛鎵浠ユ渶澶у煎湪鏌愰」绛変簬闆舵垨鍏跺墠涓椤癸紝鐢辩瓑宸暟鍒楅氶」鍏紡an=a1-(n-1)d锛屼护鍏朵负0瑙ｅ緱n=6鏃朵负a6=0锛屽垯鍓嶄簲椤圭殑鍜屼负鏈澶у糞5(鎴朣6)= 锛10+2锛*5/2 = 30
绛旓細(2)-(1)100d=-32/3 d=-32/300 S80=S50+(a51+a52+...+a80)=S50+(a1+50d+a2+50d+...+a30+50d)=S50+(a1+a2+...+a30)+30脳50d =S50+S30+5脳300脳(-32)/300 =50+30-160 =-80 濡傛灉浣犺瘉鏄庤繃涓嬮潰鐨勯鐩細绛夊樊鏁板垪锛m鈮爊锛孲m=n Sn=m锛屽垯S(m+n)=-(m+n)锛岄偅涔堝彲浠...
绛旓細璁惧叕宸负d 30=a1+(10-1)d 50=a1+(20-1)d a1=12锛宒=2 锛1锛夋暟鍒锝an锝濈殑閫氶」鍏紡涓 an=12+2(n-1)鍗砤n=2n+10 锛2锛夎嫢Sn=242 n锛12+2n+10锛/2=242 n²+11n-242=0 n=11(n=-22涓嶅悎棰樻剰鑸嶅幓)
绛旓細]-[a(m+1)+a(m+2)+鈥︹+a(2m)]=md+md+md鈥︹+md=(m^2)*d,鍗砐1,X2,X3鍛绛夊樊鏁板垪锛鐢遍鍙煡锛歑1=Sm=30,X2=S2m-Sm=70,鍒橷3=2*X2-X1=110;鈭碨3m=X1+X2+X3=210;(灏忕粨锛氱瓑宸暟鍒楁瘡杩炵画m椤逛篃鏄瓑宸暟鍒楋紝鍋氶鐔熸倝浜嗭紝閫夋嫨棰樹笉鐢ㄨ繖涔堝姝ラ鐨勶紝鐩存帴绠楀氨琛屼簡锛屽彛绠楅兘鍙互)...
