Higher Engineering Mathematics, Sixth Edition

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