Triangular Numbers

Triangular Numbers

• 
• Triangular numbers are a pattern of numbers that form equilateral triangles.
• Each number in the sequence adds a new layer of dots to each triangle.
• n equals the term in the sequence therefore this can be used to calculate how many dots are in its corresponding triangle, i.e. its triangular number;
• n=1 is the first term
• n=5 is the fifth term
• n=200 is the two hundredth term
• Caption: http://study.com/academy/lesson/what-are-triangular-numbers-definition-formula-examples.html

Slide 2

• 
• There is a pattern when counting the number of dots in the first 4 terms:
• The term and the last row of dots are the same.
• Each row has one less dot than the one ahead of it.
• Looking at the 4th term:
• n=4: 1+2+3+4 starting with 4 and counting backwards
• n=4: (4-3)+(4-2)+(4-1)+4
• To apply this to any term:
• nth term: 1+2+3+...+(n-3)+(n-2)+(n-1)+n or
• nth term: n+(n-1)+(n-2)+(n-3)+...+3+2+1
• This is ok for calculating small triangular numbers but for larger ones a formula needs to used.

Slide 3

• Formula for the nth triangle
• 1+2+3+...+n=n/2 x (n+1)
• As it's an AP with a=1, d=1

Series 1

• Notation
• ∑ notation is used to write the sum of a series in a conveniently condensed form.
• r is called the counter.
• Start with r=1 and increase its value in step of one until r=n is reached
• 

Slide 5

• 
• Caption: Note: ∑x^2 ≠ [∑x]^2 - this is a very common error

Series 2

Slide 7

• Convergence
• A sequence {Tk} tends to a Limit L if Tk becomes closer and closer to the value of L as k → ∞.
• The limit of the sequence is L and write:
• ∴ the series converges
• 

Slide 8

• Sum
• The sum Sn=T1+T2+T3+...+Tn is convergent if Sn becomes closer and closer to some limit L as n → ∞.
• Write:
• 

Slide 9

• Divergent
• A sequence/sum is divergent if it is not convergent.
• Note
• In general if Tk → 0 as k → ∞, then the sum Sn will be convergent.
• This is a necessary condition but not a sufficient condition for convergence.

Series 3

• Let
• Sn=T1+T2+T3+...+Tn
• Then
• Tn=Sn-Sn-1
• Note:
• GP = a+ar+ar^2+...
• AP = a+(a+d)+(a+2d)+...