27:
1461:
1770:
1101:. In it, uniform blocks are stacked on top of each other to achieve the maximum sideways or lateral distance covered. The blocks are stacked 1/2, 1/4, 1/6, 1/8, 1/10, … distance sideways below the original block. This ensures that the center of gravity is just at the center of the structure so that it does not collapse. A slight increase in weight on the structure causes it to become unstable and fall.
698:
1071:
of C with respect to A and B, then the distances from any one of these points to the three remaining points form harmonic progression. Specifically, each of the sequences AC, AB, AD; BC, BA, BD; CA, CD, CB; and DA, DC, DB are harmonic progressions, where each
392:
923:
506:
229:
803:
1008:
499:
256:
1652:
70:
810:
693:{\displaystyle 1,{\tfrac {\ 1\ }{2}},\ {\tfrac {\ 1\ }{3}},\ {\tfrac {\ 1\ }{4}},\ {\tfrac {\ 1\ }{5}},\ {\tfrac {\ 1\ }{6}},\ \ldots \ ,\ {\tfrac {\ 1\ }{n}},\ \ldots \ }
120:
1642:
1295:
709:
1735:
1576:
930:
1586:
1199:
1581:
1750:
1288:
1730:
1632:
1622:
1740:
1068:
1799:
1745:
1647:
1281:
1115:
1019:
1794:
1773:
1755:
1120:
387:{\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}},\ {\frac {1}{a+2d}},\ {\frac {1}{a+3d}},\cdots ,\ {\frac {1}{a+kd}},}
1627:
1617:
1607:
1637:
436:
1722:
1544:
33:
1384:
1331:
1177:
1092:
1080:
918:{\displaystyle \ -{\tfrac {30}{\ 7\ }},\ \ldots \ ,\ {\tfrac {30}{\ \left(3\ -\ 2n\right)\ }},\ \ldots \ }
97:
93:
1591:
1336:
1110:
1098:
26:
1702:
1539:
1308:
1240:
1682:
1549:
1076:
1182:
224:{\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}},\ {\frac {1}{a+2d}},\ {\frac {1}{a+3d}},\cdots ,}
1523:
1508:
1480:
1460:
1399:
1612:
1712:
1513:
1485:
1439:
1429:
1409:
1394:
1195:
1176:, Bolyai Soc. Math. Stud., vol. 25, János Bolyai Math. Soc., Budapest, pp. 289–309,
20:
1697:
1518:
1444:
1434:
1414:
1316:
1187:
1064:
1209:
1475:
1404:
1205:
798:{\displaystyle \ {\tfrac {12}{\ 5\ }},\ 2,\ \ldots \ ,\ {\tfrac {12}{\ n\ }},\ \ldots \ }
1707:
1692:
1687:
1366:
1351:
1047:. The reason is that, necessarily, at least one denominator of the progression will be
414:
247:
1788:
1672:
1346:
1267:
1169:
1146:
1032:
108:
1677:
1419:
1361:
1052:
1048:
1235:
1424:
1371:
1191:
77:
1236:
Modern geometry of the point, straight line, and circle: an elementary treatise
1153:[Generalization of an elementary number-theoretic theorem of Kürschák]
1003:{\displaystyle \ \ldots \ {\tfrac {30}{\ \left(5\ -\ 2n\right)\ }},\ \ldots \ }
114:
As a third equivalent characterization, it is an infinite sequence of the form
1356:
1150:
1304:
1125:
1072:
of the distances is signed according to a fixed orientation of the line.
1025:
89:
16:
Progression formed by taking the reciprocals of an arithmetic progression
1273:
1223:
Chapters on the modern geometry of the point, line, and circle, Vol. II
1044:
25:
1277:
1151:"Egy Kürschák-féle elemi számelméleti tétel általánosítása"
1254:
by Stan
Gibilisco, Norman H. Crowhurst, (2007) p. 221
1031:
It is not possible for a harmonic progression of distinct
944:
859:
821:
764:
717:
659:
621:
595:
569:
543:
517:
51:
1653:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
1643:
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
933:
813:
712:
509:
439:
259:
123:
36:
1097:
An excellent example of
Harmonic Progression is the
1721:
1665:
1600:
1569:
1562:
1532:
1501:
1494:
1468:
1380:
1324:
1315:
1002:
917:
797:
692:
493:
386:
223:
64:
30:The first ten members of the harmonic sequence
1083:, then the sides are in harmonic progression.
1289:
1260:by Chemical Rubber Company (1974) p. 102
1172:(2013), "Paul Erdős and Egyptian fractions",
494:{\displaystyle \ n=1,\ 2,\ 3,\ 4,\ \ldots \ }
8:
1736:Hypergeometric function of a matrix argument
1055:that does not divide any other denominator.
1592:1 + 1/2 + 1/3 + ... (Riemann zeta function)
1264:Essentials of algebra for secondary schools
1566:
1498:
1321:
1296:
1282:
1274:
1648:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
1181:
943:
932:
858:
820:
812:
763:
716:
711:
658:
620:
594:
568:
542:
516:
508:
438:
360:
327:
300:
276:
260:
258:
191:
164:
140:
124:
122:
50:
41:
35:
1138:
1024:Infinite harmonic progressions are not
1067:A, B, C, and D are such that D is the
65:{\displaystyle a_{n}={\tfrac {1}{n}}}
7:
1613:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
1035:(other than the trivial case where
250:, or a finite sequence of the form
433:is a natural number, in sequence:
105:sequence is a harmonic progression
14:
1731:Generalized hypergeometric series
1769:
1768:
1741:Lauricella hypergeometric series
1459:
1225:by Richard Townsend (1865) p. 24
1751:Riemann's differential equation
1252:Mastering Technical Mathematics
1:
1746:Modular hypergeometric series
1587:1/4 + 1/16 + 1/64 + 1/256 + ⋯
1020:Harmonic series (mathematics)
1014:Sums of harmonic progressions
1258:Standard mathematical tables
1756:Theta hypergeometric series
1192:10.1007/978-3-642-39286-3_9
1121:List of sums of reciprocals
1816:
1638:Infinite arithmetic series
1582:1/2 + 1/4 + 1/8 + 1/16 + ⋯
1577:1/2 − 1/4 + 1/8 − 1/16 + ⋯
1090:
1017:
111:of the neighboring terms.
19:For the musical term, see
18:
1764:
1457:
405:is a natural number and −
1469:Properties of sequences
1332:Arithmetic progression
1093:Block-stacking problem
1081:arithmetic progression
1075:In a triangle, if the
1004:
927:10, 30, −30, −10, −6,
919:
799:
694:
495:
388:
225:
107:when each term is the
98:arithmetic progression
73:
66:
1723:Hypergeometric series
1337:Geometric progression
1111:Geometric progression
1099:Leaning Tower of Lire
1087:Leaning Tower of Lire
1005:
920:
800:
695:
496:
389:
226:
92:formed by taking the
67:
29:
1800:Sequences and series
1703:Trigonometric series
1495:Properties of series
1342:Harmonic progression
1241:John Alexander Third
931:
811:
710:
507:
437:
257:
121:
82:harmonic progression
34:
1795:Mathematical series
1683:Formal power series
1028:(sum to infinity).
417:or is greater than
1481:Monotonic function
1400:Fibonacci sequence
1270:(1897) p. 307
1069:harmonic conjugate
1043:= 0) to sum to an
1000:
986:
915:
901:
838:
807:30, −30, −10, −6,
795:
781:
734:
690:
676:
638:
612:
586:
560:
534:
491:
384:
221:
74:
62:
60:
1782:
1781:
1713:Generating series
1661:
1660:
1633:1 − 2 + 4 − 8 + ⋯
1628:1 + 2 + 4 + 8 + ⋯
1623:1 − 2 + 3 − 4 + ⋯
1618:1 + 2 + 3 + 4 + ⋯
1608:1 + 1 + 1 + 1 + ⋯
1558:
1557:
1486:Periodic sequence
1455:
1454:
1440:Triangular number
1430:Pentagonal number
1410:Heptagonal number
1395:Complete sequence
1317:Integer sequences
1201:978-3-642-39285-6
1170:Graham, Ronald L.
999:
993:
985:
983:
969:
963:
952:
942:
936:
914:
908:
900:
898:
884:
878:
867:
857:
851:
845:
837:
835:
829:
816:
794:
788:
780:
778:
772:
762:
756:
750:
741:
733:
731:
725:
715:
702:harmonic sequence
689:
683:
675:
670:
664:
657:
651:
645:
637:
632:
626:
619:
611:
606:
600:
593:
585:
580:
574:
567:
559:
554:
548:
541:
533:
528:
522:
490:
484:
475:
466:
457:
442:
429:In the following
379:
359:
346:
326:
319:
299:
292:
275:
268:
238:is not zero and −
210:
190:
183:
163:
156:
139:
132:
86:harmonic sequence
59:
21:Chord progression
1807:
1772:
1771:
1698:Dirichlet series
1567:
1499:
1463:
1435:Polygonal number
1415:Hexagonal number
1388:
1322:
1298:
1291:
1284:
1275:
1244:
1232:
1226:
1220:
1214:
1212:
1185:
1174:Erdős centennial
1166:
1161:(in Hungarian),
1156:
1143:
1065:collinear points
1009:
1007:
1006:
1001:
997:
991:
987:
984:
981:
980:
976:
967:
961:
950:
945:
940:
934:
924:
922:
921:
916:
912:
906:
902:
899:
896:
895:
891:
882:
876:
865:
860:
855:
849:
843:
839:
836:
833:
827:
822:
814:
804:
802:
801:
796:
792:
786:
782:
779:
776:
770:
765:
760:
754:
748:
739:
735:
732:
729:
723:
718:
713:
699:
697:
696:
691:
687:
681:
677:
671:
668:
662:
660:
655:
649:
643:
639:
633:
630:
624:
622:
617:
613:
607:
604:
598:
596:
591:
587:
581:
578:
572:
570:
565:
561:
555:
552:
546:
544:
539:
535:
529:
526:
520:
518:
500:
498:
497:
492:
488:
482:
473:
464:
455:
440:
432:
393:
391:
390:
385:
380:
378:
361:
357:
347:
345:
328:
324:
320:
318:
301:
297:
293:
291:
277:
273:
269:
261:
230:
228:
227:
222:
211:
209:
192:
188:
184:
182:
165:
161:
157:
155:
141:
137:
133:
125:
103:Equivalently, a
71:
69:
68:
63:
61:
52:
46:
45:
1815:
1814:
1810:
1809:
1808:
1806:
1805:
1804:
1785:
1784:
1783:
1778:
1760:
1717:
1666:Kinds of series
1657:
1596:
1563:Explicit series
1554:
1528:
1490:
1476:Cauchy sequence
1464:
1451:
1405:Figurate number
1382:
1376:
1367:Powers of three
1311:
1302:
1248:
1247:
1233:
1229:
1221:
1217:
1202:
1168:
1159:Mat. Fiz. Lapok
1154:
1145:
1144:
1140:
1135:
1116:Harmonic series
1107:
1095:
1089:
1061:
1059:Use in geometry
1022:
1016:
957:
953:
949:
929:
928:
872:
868:
864:
826:
809:
808:
769:
722:
708:
707:
661:
623:
597:
571:
545:
519:
505:
504:
435:
434:
430:
427:
365:
332:
305:
281:
255:
254:
196:
169:
145:
119:
118:
37:
32:
31:
24:
17:
12:
11:
5:
1813:
1811:
1803:
1802:
1797:
1787:
1786:
1780:
1779:
1777:
1776:
1765:
1762:
1761:
1759:
1758:
1753:
1748:
1743:
1738:
1733:
1727:
1725:
1719:
1718:
1716:
1715:
1710:
1708:Fourier series
1705:
1700:
1695:
1693:Puiseux series
1690:
1688:Laurent series
1685:
1680:
1675:
1669:
1667:
1663:
1662:
1659:
1658:
1656:
1655:
1650:
1645:
1640:
1635:
1630:
1625:
1620:
1615:
1610:
1604:
1602:
1598:
1597:
1595:
1594:
1589:
1584:
1579:
1573:
1571:
1564:
1560:
1559:
1556:
1555:
1553:
1552:
1547:
1542:
1536:
1534:
1530:
1529:
1527:
1526:
1521:
1516:
1511:
1505:
1503:
1496:
1492:
1491:
1489:
1488:
1483:
1478:
1472:
1470:
1466:
1465:
1458:
1456:
1453:
1452:
1450:
1449:
1448:
1447:
1437:
1432:
1427:
1422:
1417:
1412:
1407:
1402:
1397:
1391:
1389:
1378:
1377:
1375:
1374:
1369:
1364:
1359:
1354:
1349:
1344:
1339:
1334:
1328:
1326:
1319:
1313:
1312:
1303:
1301:
1300:
1293:
1286:
1278:
1272:
1271:
1261:
1255:
1246:
1245:
1227:
1215:
1200:
1167:. As cited by
1137:
1136:
1134:
1131:
1130:
1129:
1123:
1118:
1113:
1106:
1103:
1091:Main article:
1088:
1085:
1060:
1057:
1033:unit fractions
1018:Main article:
1015:
1012:
1011:
1010:
996:
990:
979:
975:
972:
966:
960:
956:
948:
939:
925:
911:
905:
894:
890:
887:
881:
875:
871:
863:
854:
848:
842:
832:
825:
819:
805:
791:
785:
775:
768:
759:
753:
747:
744:
738:
728:
721:
704:
700:is called the
686:
680:
674:
667:
654:
648:
642:
636:
629:
616:
610:
603:
590:
584:
577:
564:
558:
551:
538:
532:
525:
515:
512:
487:
481:
478:
472:
469:
463:
460:
454:
451:
448:
445:
426:
423:
415:natural number
395:
394:
383:
377:
374:
371:
368:
364:
356:
353:
350:
344:
341:
338:
335:
331:
323:
317:
314:
311:
308:
304:
296:
290:
287:
284:
280:
272:
267:
264:
248:natural number
232:
231:
220:
217:
214:
208:
205:
202:
199:
195:
187:
181:
178:
175:
172:
168:
160:
154:
151:
148:
144:
136:
131:
128:
58:
55:
49:
44:
40:
15:
13:
10:
9:
6:
4:
3:
2:
1812:
1801:
1798:
1796:
1793:
1792:
1790:
1775:
1767:
1766:
1763:
1757:
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1732:
1729:
1728:
1726:
1724:
1720:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1673:Taylor series
1671:
1670:
1668:
1664:
1654:
1651:
1649:
1646:
1644:
1641:
1639:
1636:
1634:
1631:
1629:
1626:
1624:
1621:
1619:
1616:
1614:
1611:
1609:
1606:
1605:
1603:
1599:
1593:
1590:
1588:
1585:
1583:
1580:
1578:
1575:
1574:
1572:
1568:
1565:
1561:
1551:
1548:
1546:
1543:
1541:
1538:
1537:
1535:
1531:
1525:
1522:
1520:
1517:
1515:
1512:
1510:
1507:
1506:
1504:
1500:
1497:
1493:
1487:
1484:
1482:
1479:
1477:
1474:
1473:
1471:
1467:
1462:
1446:
1443:
1442:
1441:
1438:
1436:
1433:
1431:
1428:
1426:
1423:
1421:
1418:
1416:
1413:
1411:
1408:
1406:
1403:
1401:
1398:
1396:
1393:
1392:
1390:
1386:
1379:
1373:
1370:
1368:
1365:
1363:
1362:Powers of two
1360:
1358:
1355:
1353:
1350:
1348:
1347:Square number
1345:
1343:
1340:
1338:
1335:
1333:
1330:
1329:
1327:
1323:
1320:
1318:
1314:
1310:
1306:
1299:
1294:
1292:
1287:
1285:
1280:
1279:
1276:
1269:
1268:Webster Wells
1265:
1262:
1259:
1256:
1253:
1250:
1249:
1242:
1238:
1237:
1231:
1228:
1224:
1219:
1216:
1211:
1207:
1203:
1197:
1193:
1189:
1184:
1183:10.1.1.300.91
1179:
1175:
1171:
1164:
1160:
1152:
1148:
1142:
1139:
1132:
1127:
1124:
1122:
1119:
1117:
1114:
1112:
1109:
1108:
1104:
1102:
1100:
1094:
1086:
1084:
1082:
1078:
1073:
1070:
1066:
1058:
1056:
1054:
1050:
1046:
1042:
1038:
1034:
1029:
1027:
1021:
1013:
994:
988:
977:
973:
970:
964:
958:
954:
946:
937:
926:
909:
903:
892:
888:
885:
879:
873:
869:
861:
852:
846:
840:
830:
823:
817:
806:
789:
783:
773:
766:
757:
751:
745:
742:
736:
726:
719:
706:12, 6, 4, 3,
705:
703:
684:
678:
672:
665:
652:
646:
640:
634:
627:
614:
608:
601:
588:
582:
575:
562:
556:
549:
536:
530:
523:
513:
510:
503:
502:
501:
485:
479:
476:
470:
467:
461:
458:
452:
449:
446:
443:
424:
422:
420:
416:
412:
408:
404:
401:is not zero,
400:
381:
375:
372:
369:
366:
362:
354:
351:
348:
342:
339:
336:
333:
329:
321:
315:
312:
309:
306:
302:
294:
288:
285:
282:
278:
270:
265:
262:
253:
252:
251:
249:
245:
241:
237:
218:
215:
212:
206:
203:
200:
197:
193:
185:
179:
176:
173:
170:
166:
158:
152:
149:
146:
142:
134:
129:
126:
117:
116:
115:
112:
110:
109:harmonic mean
106:
101:
99:
95:
91:
87:
83:
79:
56:
53:
47:
42:
38:
28:
22:
1678:Power series
1420:Lucas number
1372:Powers of 10
1352:Cubic number
1341:
1263:
1257:
1251:
1243:(1898) p. 44
1234:
1230:
1222:
1218:
1173:
1162:
1158:
1141:
1096:
1074:
1062:
1053:prime number
1040:
1036:
1030:
1023:
701:
428:
418:
410:
406:
402:
398:
396:
243:
239:
235:
233:
113:
104:
102:
85:
81:
75:
1545:Conditional
1533:Convergence
1524:Telescoping
1509:Alternating
1425:Pell number
94:reciprocals
90:progression
78:mathematics
1789:Categories
1570:Convergent
1514:Convergent
1133:References
1128:(in music)
1601:Divergent
1519:Divergent
1381:Advanced
1357:Factorial
1305:Sequences
1178:CiteSeerX
1147:Erdős, P.
1126:Harmonics
1077:altitudes
1049:divisible
995:…
965:−
938:…
910:…
880:−
847:…
818:−
790:…
752:…
685:…
647:…
486:…
413:is not a
352:⋯
246:is not a
216:⋯
1774:Category
1540:Absolute
1149:(1932),
1105:See also
1039:= 1 and
1026:summable
425:Examples
1550:Uniform
1210:3203600
1165:: 17–24
1079:are in
1045:integer
88:) is a
1502:Series
1309:series
1208:
1198:
1180:
998:
992:
982:
968:
962:
951:
941:
935:
913:
907:
897:
883:
877:
866:
856:
850:
844:
834:
828:
815:
793:
787:
777:
771:
761:
755:
749:
740:
730:
724:
714:
688:
682:
669:
663:
656:
650:
644:
631:
625:
618:
605:
599:
592:
579:
573:
566:
553:
547:
540:
527:
521:
489:
483:
474:
465:
456:
441:
397:where
358:
325:
298:
274:
234:where
189:
162:
138:
96:of an
1445:array
1325:Basic
1155:(PDF)
1051:by a
1385:list
1307:and
1196:ISBN
84:(or
80:, a
1266:by
1239:by
1188:doi
1063:If
76:In
1791::
1206:MR
1204:,
1194:,
1186:,
1163:39
1157:,
947:30
862:30
824:30
767:12
720:12
431:n
421:.
100:.
1387:)
1383:(
1297:e
1290:t
1283:v
1213:.
1190::
1041:k
1037:a
989:,
978:)
974:n
971:2
959:5
955:(
904:,
893:)
889:n
886:2
874:3
870:(
853:,
841:,
831:7
784:,
774:n
758:,
746:,
743:2
737:,
727:5
679:,
673:n
666:1
653:,
641:,
635:6
628:1
615:,
609:5
602:1
589:,
583:4
576:1
563:,
557:3
550:1
537:,
531:2
524:1
514:,
511:1
480:,
477:4
471:,
468:3
462:,
459:2
453:,
450:1
447:=
444:n
419:k
411:d
409:/
407:a
403:k
399:a
382:,
376:d
373:k
370:+
367:a
363:1
355:,
349:,
343:d
340:3
337:+
334:a
330:1
322:,
316:d
313:2
310:+
307:a
303:1
295:,
289:d
286:+
283:a
279:1
271:,
266:a
263:1
244:d
242:/
240:a
236:a
219:,
213:,
207:d
204:3
201:+
198:a
194:1
186:,
180:d
177:2
174:+
171:a
167:1
159:,
153:d
150:+
147:a
143:1
135:,
130:a
127:1
72:.
57:n
54:1
48:=
43:n
39:a
23:.
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