five pentagonal numbers that follow 22
Pn = (3n^2-n)/2
To find the pentagonal numbers that follow 22, let's first understand what pentagonal numbers are.
Pentagonal numbers are a sequence of numbers that can be represented by dots arranged in the shape of a pentagon. The formula to find the nth pentagonal number is given by:
Pn = n(3n - 1) / 2
Now, let's find the pentagonal numbers that follow 22:
1. Start with the given number, 22.
2. Calculate the value of n for the next pentagonal number by using the formula n = ((sqrt(24x + 1) + 1) / 6). In this case, x = 22.
n = ((sqrt(24x + 1) + 1) / 6)
n = ((sqrt(24*22 + 1) + 1) / 6)
n ≈ 4.273
Since n should be an integer, we can round it up to the nearest whole number, which gives us n = 5.
3. Substitute the value of n = 5 into the pentagonal number formula to find the next pentagonal number:
P5 = 5(3*5 - 1) / 2
P5 = 5(15 - 1) / 2
P5 = 5(14) / 2
P5 = 35
Therefore, the five pentagonal numbers that follow 22 are: 35, 54, 77, 104, and 135.