Experience

Experience, commonly abbreviated as exp or xp, is a measure of progress in a certain skill. It is obtained by performing tasks related to that skill. Experience can also sometimes be gained by performing certain other tasks not necessarily related to the skill, such as completing quests, receiving the Book of Knowledge from the Surprise Exam random event, receiving a lamp from the genie random event, doing certain mini-games, getting a lamp from Cap'n Izzy No-Beard for completing a part of the Karamja Achievement Diary, etc. After gaining a certain amount of experience, players will advance to the next level in that skill, which can result in new abilities, amongst other things. The amount of experience needed for the next level increases by approximately 10% each level. For example, 83 experience is required for advancement to level 2, while 91 experience is required for advancement to level 3.

Below is an experience table showing the amount of total experience needed for each level. The experience difference shows the amount of experience between the previous level and the level. Note that a 10% growth factor may seem slow, but, as with all exponential growth, it expands rapidly to a massive 13,034,431 experience needed for level 99. Level 85 requires nearly one quarter of the experience needed for Level 99 and Level 92 is nearly the exact halfway mark, requiring 6,517,253 experience. This clearly demonstrates the experience gap that grows rapidly at higher levels.

Each skill has a maximum amount of xp that you can receive at 200,000,000. After you reach this point, you can continue doing that skill; however, you will not receive any more xp.

Levels 1-99
The experience difference calculation is a function in the form of $$f(x) = \frac{\lfloor x-1+300 * 2^{\frac{x-1}{7}} \rfloor}{4}$$, x being the level starting from 2.

Formula
The formula needed to calculate the amount of experience needed to reach level L is:

$$\text{Experience} = \left \lfloor{\frac{1}{4}\sum_{x=1}^{L-1}}\left\lfloor{x + 300\cdot2^{\frac{x}{7}}}\right\rfloor\right\rfloor$$

If the floor functions are ignored, the sum is merely a combination of arithmetic and geometric sums, and can thus be further derived into a closed-form continuous function with a positive real variable L (standing for level). The approximation rapidly approaches the original sum asymptotically. The function is as follows:

$$\frac{1}{4} \left[ \frac{1}{2} \cdot {L}^{2} - \frac{1}{2} \cdot L + 300 \cdot \sqrt[7]{2} \cdot \frac{{2}^{\frac{L-1}{7}}-1} {{2}^{\frac{1}{7}}-1} \right] + E \left( L \right)$$

Ignoring floor functions will result in an exceedingly minor error term, and as floor of real numbers is taken twice, once from the inner function of sum and once from the total sum, the error term E(L) is strictly bounded:

$$0 \leq E\left ( L \right ) \leq L + 1$$

According to these bounds the maximum error at level 99 would only be 100, whereas the actual error, 9.97, is even lower. The aforementioned bounds can be improved, although it is redundant with this low errors.

Graphs
The experience required for each level can be graphed. Graphing the experience required on a linear scale shows that the experience required is essentially exponential. Graphing the same data on a logarithmic scale shows that the function deviates from being exponential before around level 10.

Trivia
Due to the exponential nature of experience needed for each level (for levels above 10): After level 30, the general experience required to level up falls between 9-10% of your current experience. At mid to high levels of 55+, the experience needed to level up falls at around an average of 9.4% of current experience all the way to level 99.
 * to level up 9 more times after levelling up once, it will take 15.25 times the xp needed for the initial level up.
 * similarly for 19 more level ups, it will take 59 times the initial xp.
 * similarly for 29 more level ups, it will take 176.768 times the initial xp. formula
 * the xp needed to level up 11 times is not enough to level up even once 30 levels from now. formula
 * After about level 20, where the exponential experience curve fits the best, the amount of total experience doubles every 7 levels. (i.e. Level 85 requires twice as much total experience as level 78, level 92 requires twice as much total experience as level 85, and level 99 requires twice as much total experience as level 92.)

To max all of the 13 f2p skills it will take a total of 169,447,603 experience. To reach the 200 million experience cap for all the 13 f2p skills it will take a total of 2,600,000,000 experience. To max all of the 21 p2p skills it will take a total of 273,723,051 experience. To reach the 200 million experience cap for all the 21 p2p skills it will take a total of 4,200,000,000 experience.