I am trying to derive a formula for this sequence:
3, 4, 6, 8, 12, ...
Need Help!!
one method:
define the sequence as a function of $\displaystyle n$, where $\displaystyle n \ge 1$ is an integer.
so you have 5 points. (1,3), (2,4), (3,6), (4,8), and (5,12). now we want to find a curve that passes through all those points. let the curve be given by:
$\displaystyle f(n) = an^4 + bn^3 + cn^2 + dn + e$
we have:
$\displaystyle f(1) = a + b + c + d + e = 3$ ..............(1)
$\displaystyle f(2) = 16a + 8b + 4c + 2d + e = 4$ ..............(2)
$\displaystyle f(3) = 81a + 27b + 9c + 3d + e = 6$ .............(3)
$\displaystyle f(4) = 256a + 64b + 16c + 4b + e = 8$.........(4)
$\displaystyle f(5) = 625a + 125b + 25c + 5d + e = 12$ ..........(5)
you have 5 equations, 5 unknowns. solve simultaneously