660 Higher Engineering Mathematics
Arithmetic progression:
Ifa=first term andd=common difference, then the
arithmetic progression is:a,a+d,a+ 2 d,...
Then’th term is:a+(n− 1 )d
Sum ofnterms,Sn=
n
2
[2a+(n− 1 )d]
Geometric progression:
Ifa=first term andr=common ratio, then the geomet-
ric progression is:a,ar,ar^2 ,...
Then’th term is:arn−^1
Sum ofnterms,Sn=
a( 1 −rn)
( 1 −r)
or
a(rn− 1 )
(r− 1 )
If− 1 <r<1,S∞=
a
( 1 −r)
Binomial series:
(a+b)n=an+nan−^1 b+
n(n− 1 )
2!
an−^2 b^2
+
n(n− 1 )(n− 2 )
3!
an−^3 b^3 +···
( 1 +x)n= 1 +nx+
n(n− 1 )
2!
x^2
+
n(n− 1 )(n− 2 )
3!
x^3 +···
Maclaurin’s series:
f(x)=f( 0 )+xf′( 0 )+
x^2
2!
f′′( 0 )
+
x^3
3!
f′′′( 0 )+···
Newton Raphson iterative method:
Ifr 1 is the approximate value for a real root of the equa-
tionf(x)=0, then a closer approximation to the root,
r 2 , is given by:
r 2 =r 1 −
f(r 1 )
f′(r 1 )
Boolean algebra:
Laws and rules of Boolean algebra
Commutative Laws: A+B=B+A
A·B=B·A
Associative Laws: A+B+C=(A+B)+C
A·B·C=(A·B)·C
Distributive Laws: A·(B+C)=A·B+A·C
A+(B·C)=(A+B)·(A+C)
Sum rules: A+A= 1
A+ 1 = 1
A+ 0 =A
A+A=A
Product rules: A·A= 0
A· 0 = 0
A· 1 =A
A·A=A
Absorption rules: A+A·B=A
A·(A+B)=A
A+A·B=A+B
De Morgan’s Laws: A+B=A·B
A·B=A+B
Geometry and Trigonometry
Theorem of Pythagoras:
b^2 =a^2 +c^2
A
B C
c
b
a
Figure FA1
Identities:
secθ=
1
cosθ
cosecθ=
1
sinθ
cotθ=
1
tanθ
tanθ =
sinθ
cosθ
cos^2 θ+sin^2 θ= 11 +tan^2 θ=sec^2 θ
cot^2 θ+ 1 =cosec^2 θ