Triangular Numbers

Triangular Numbers

Triangular Numbers (image/jpeg)

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 termn=5 is the fifth termn=200 is the two hundredth term

Caption: : http://study.com/academy/lesson/what-are-triangular-numbers-definition-formula-examples.html

Triangular Number Table (image/jpeg)

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 backwardsn=4: (4-3)+(4-2)+(4-1)+4To apply this to any term:nth term: 1+2+3+...+(n-3)+(n-2)+(n-1)+n  ornth term: n+(n-1)+(n-2)+(n-3)+...+3+2+1This is ok for calculating small triangular numbers but for larger ones a formula needs to used.

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

Tri Num 1 (image/png)

Tri Num 2 (image/png)

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

Series 2

Tri Num 3 (image/png)

ConvergenceA 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

Tri Num 4 (image/png)

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

Tri Num 5 (image/png)

DivergentA sequence/sum is divergent if it is not convergent.NoteIn 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)+...