FM for Actuaries

328 APPENDIX A
A.3 Roots of a quadratic equation
The following quadratic equation in x
has the following two roots
ax 2 + bx + c =0
−b ± √ b 2 − 4ac
.
2a
If b 2 − 4ac < 0, the equation has two complex roots.
A.4 Arithmetic progression
Consider the sum of the following arithmetic progression with n terms
S = a +(a + d)+(a +2d)+···+(a +(n − 1)d),
with initial value a and common difference d. ThevalueofS is
na +
n(n − 1)
d.
2
A.5 Geometric progression
Consider the sum of the following geometric progression with n terms
S = a + ar + ar 2 + ···+ ar n−1 ,
with initial value a andcommonratior. ThevalueofS is
a(1 − r n )
.
1 − r
If |r| < 1, thenr n → 0 as n →∞,sothatS → a/(1 − r) as n →∞.
A.6 Some derivatives
dx n
dx = nxn−1 ,
da x
dx = ax ln a,
d ln x
dx = 1 x .