,
Sequences & Series : geometrical series
 

[ structure ][ sum proof ][ mean ][sum to infinity]

 

 

 

 

Geometrical series structure

A geometricall series starts with the first term, usually given the letter 'a'. For each subsequent term of the series the first term is multiplied by another term. The term is a multiple of the letter 'r' called 'the common ratio '.

So the series has the structure:

geometrical series structure

where Snis the sum to 'n' terms, the letter 'l' is the last term.

The common ratio 'r' is calculated by dividing any term by the term before it.

The nth term(sometimes called the 'general term')is given by:

geometrical series general term

 

back to top

 

Proof of the sum of a geometrical series

geometrical series sum

NB an alternative formula for r > 1 , just multiply numerator & denominator by -1

Example #1

In a geometrical progression the sum of the 3rd & 4th terms is 60 and the sum of the 4th & 5th terms is 120.

Find the 1st term and the common ratio.

geometrical series problem#1

 

back to top

 

Example #2

What is the smallest number of terms of the geometrical progression

2 + 6 + 18 + 54 + 162 ...

that will give a total greater than 1000?

geometrical series problem#2

Geometic Mean

This is a method of finding a term sandwiched between two other terms.

So if we have a sequence of terms: a b c and a and c are known. The ratio of successive terms gives the common ratio. Equating these:

geometrical series mean

Example

If the 4th term of a geometrical progression is 40 and the 6th is 160, what is the 5th term?

geometrical series problem#3

 

back to top

 

Sum to infinity

This concerns geometrical progressions that as the number of terms increase, the value of the sum approaches one specific number. This number is called the sum to infinity.

In this example as 'n' increases the sum approaches 2.

eg of infinite geometrical series

infinite GM #2

So if the term rn tends to zero, with increasing n the equation for the sum to n terms changes:

GM sum changed

Example

Express 0.055555... as a fraction.

 

infinite GP problem#1

 

 

 

back to top

 

your stop for the best in math, science & programming tutorials on the Net revision help to get a better result incremental success advanced physics for secondary/high school, including much in-depth content common to first year university courses your one stop for the best in math, science and programming tuition revision help for a better result incremental success advanced physics for high school/secondary and 1st year university fast-track learning for everyone

[ PURE MATHS ][ MECHANICS ][ STATISTICS ]

VIDEO

parametric differen.
equation of a tangent
equation of a normal
rate of change prob.1
rate of change prob.2
the Chain Rule
Chain Rule probs. #1
Chain Rule probs. #2
Chain Rule probs. #3
the Product Rule
parametric eqs.prob#1
parametric eqs.prob#2
intro. to integration
integration by parts 1
integration by parts 2
area under a curve
volumes of revolution
area between curves
Binomial Theorem
Bin. Theorem problems
Trig. Identities
Half Angle Formula
Double Angle Formula
Vectors,mod,resultant
Vector problems
Vector problems in 3D
MORE . . .
 

INTERACTIVE

 
derivative formula
tangents & normals
inverse functions
MORE . . .
 

EXAM PAPERS(.pdf)

 
Edxl C1 Pure specimen
Edxl C1 Pure answers
Edxl C2 Pure specimen
Edxl C2 Pure answers
Edxl C3 Pure specimen
Edxl C3 Pure answers
Edxl C4 Pure specimen
Edxl C4 Pure answers
MORE . . .

TOPIC NOTES(.pdf)

the derivative formula
tangents & normals
maxima & minima
chain rule
diffn.exponentials,logs
diffn.trigonometric fns.
product rule
quotient rule
parametric equations
implicit equations
differential equations
integration formula
int.'by substitution'
int.'by parts'
algebraic fractions
definite integrals
areas under curves
volumes of revolution
Trapezium Rule
integ. diff. equations
radians
sine, cosine, tangent
sec, cosec, cotan
Sine & Cosine Rules
Pythagorean ID's
compound angles
Sigma Notation
arithmetic progression
geometic progression
line between points
more straight lines
parametric equations
circles & ellipses
MORE . . .
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Google