Summation chart

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Bonus (n) Ranks Needed (r)
Seed (σ)
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

The summation chart is used to determine the number of ranks required to attain a bonus, based on an incremental additive. It is used particularly for Lore bonuses in certain spells.

For spells that provide a summation bonus, the first step to determine the progression at which you will gain further benefits is to find the first rank at which you gain a bonus. This is commonly called the "seed" (σ). Find this rank number in the red row underneath the red Seed heading. Since the seed is also the first rank at which you gain a bonus, it corresponds with '1' under the Bonus column.

Now, reading downward from the seed will tell you the next rank at which the next bonus can be achieved. Each of these milestones corresponds to their respective numbers in the Bonus column.

The method is based on triangular numbers and allows for a relatively straight forward manner to set the skill required corresponding to a given bonus threshold simultaneously to the degree with which the returns diminish.

Example

The Lore Benefits section of the spell page for Spirit Warding II (107) reads as follows:

Spiritual Lore, Blessings provides a chance to temporarily experience a +25 boost in a warding attempt based on a seed 10 summation of ranks.

The Seed in this case is 10, which corresponds to a Bonus of 1 in the chart, or a 1% chance for activation. Tracing downward from the seed shows that the next bonus is at 21 ranks, or 2% chance for activation. The progression continues in this manner until it maximizes at 186 ranks for pures (12% chance for activation) under normal training ability, or 14% chance for activation with 231 ranks made possible using enhancive items. Bonuses occurring at seed 10 are intended by design to be relatively rare occurrences.

Lore charts have been added to spell pages to illustrate bonuses.

Spiritual Lore, Blessings ranks 10 21 33 46 60 75 91 108 126 145 165 186 208 231
Chance for activation 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14%

Calculation

Bonus n = ⌊1 - σ + √(σ - 1)2 + 2r
Skill Ranks Needed r = (n2 - n)/2 + nσ
Seed σ = (1 - n)/2 + r/n

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