Summation chart

A chart used to determine the number of ranks required to attain a bonus, based on an incremental additive scale. It is used particularly for Lore bonuses in certain spells. To use the chart all you need to know is the number of skill ranks (not skill bonus) needed to first see a bonus. This value is known as the Seed.

Calculation

X	Bonus
Y	Seed
Z	Skill Ranks Needed

 X = (sqrt(4Y^2 - 4Y + 8Z + 1) - 2Y + 1) / 2

 Y = -(X^2 - X - 2Z) / 2X

 Z = (X^2 - X) / 2 + XY

Summation Chart

Bonus (X)	Seed (Y)										Ranks Needed (Z)
1	1	2	3	4	5	6	7	8	9	10
2	3	5	7	9	11	13	15	17	19	21
3	6	9	12	15	18	21	24	27	30	33
4	10	14	18	22	26	30	34	38	42	46
5	15	20	25	30	35	40	45	50	55	60
6	21	27	33	39	45	51	57	63	69	75
7	28	35	42	49	56	63	70	77	84	91
8	36	44	52	60	68	76	84	92	100	108
9	45	54	63	72	81	90	99	108	117	126
10	55	65	75	85	95	105	115	125	135	145
11	66	77	88	99	110	121	132	143	154	165
12	78	90	102	114	126	138	150	162	174	186
13	91	104	117	130	143	156	169	182	195	208
14	105	119	133	147	161	175	189	203	217	231
15	120	135	150	165	180	195	210	225	240	255
16	136	152	168	184	200	216	232	248	264	280
17	153	170	187	204	221	238	255	272	289
18	171	189	207	225	243	261	279	297
19	190	209	228	247	266	285
20	210	230	250	270	290
21	231	252	273	294
22	253	275	297
23	276	299
24	300

Resources

http://www.play.net/gs4/info/spells/lores.asp#Summation - The summation chart