Pentagonal Number
The nth pentagonal figurate number.
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Example
P_3 = 12
How it works
Pentagonal numbers count dots arranged so that each new layer extends a pentagon-shaped figure. The formula P_n = n(3n-1)/2 comes from adding the number of dots in successive layers: the first few increments grow by 3 more dots each time, which produces a quadratic pattern. This is why pentagonal numbers are a special figurate sequence rather than an arbitrary arithmetic rule.
When to use it
Use this formula when you need the number of dots in the nth pentagonal pattern or when working with figurate number sequences in number theory. It is also useful for recognizing whether a value could belong to the pentagonal-number sequence.
Worked example, step by step
Find the 5th pentagonal number. P_5 = 5(3·5 - 1)/2 P_5 = 5(15 - 1)/2 P_5 = 5·14/2 P_5 = 70/2 P_5 = 35
Common mistakes to avoid
- Using n^2 instead of n(3n-1)/2
- Forgetting to subtract 1 before dividing by 2
- Confusing pentagonal numbers with triangular or hexagonal numbers
Did You Know?
Pentagonal numbers surface unexpectedly in Euler's formula for counting the ways to add up whole numbers.
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