# hard Sequence and Series Question!! (1 Viewer)

#### kevda1st

##### Member

1,3,6,10,15....are called triangular numbers

Use the formula for the sum of an arithmetic series to find the 30th triangular number.

#### MathsTutorz

##### New Member

1,3,6,10,15....are called triangular numbers

Use the formula for the sum of an arithmetic series to find the 30th triangular number.
1 = 1
3 = 1+2
6 = 1 + 2 + 3
10 = 1 + 2 + 3 + 4

30th number will be

30/2 (2 + 29 x 1 ) = 465

#### ghostman_on_3rd

##### Member
1, 3, 6,10,15,.......

New set of series you can make from top
1+2+3+4+5+6+......
true since 1+2 = T2 from original
1+2+3 = T3 from original
1+2+3+4 = T4 (" ")
so on....

a= 1 n=30 d= 1

s30= 30/2 (2x1+(30-1)x1)
= 15x31
= 465

lol kev, cya

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#### orwellstudent

##### New Member
1, 3, 6,10,15,.......

New set of series you can make from top
1+2+3+4+5+6+......
true since 1+2 = T2 from original
1+2+3 = T3 from original
1+2+3+4 = T4 (" ")
so on....

a= 1 n=30 d= 1

s30= 30/2 (2x1+(30-1)x1)
= 15x31
= 465

lol kev, cya
Sorry, I know this was a post a LOOONNGG time ago but why is d=1? Though you did say you made a replica series with 1+2+3+4+5... but may I ask of how exactly you derived that? If anyone passing this knows, please reach out. I'm sorry if it's a intuitive question, I just don't get the reasoning behind it yet. Thanks!

#### InteGrand

##### Well-Known Member
Sorry, I know this was a post a LOOONNGG time ago but why is d=1? Though you did say you made a replica series with 1+2+3+4+5... but may I ask of how exactly you derived that? If anyone passing this knows, please reach out. I'm sorry if it's a intuitive question, I just don't get the reasoning behind it yet. Thanks!
$\bg_white \noindent It (seeing that the n-th term of the sequence is 1 + 2 + \cdots + n, for n\geq 1) is really just by inspection. You can also see it by actually drawing triangles'' (hence the name \emph{triangular numbers}''), as follows:$

(1 layer) T1 = 1:

(2 layers) T2 = 3:

••

(3 layers) T3 = 6:

••
•••

(4 layers) T4 = 10:

••
•••
••••

(5 layers) T5 = 15:

••
•••
••••
•••••,

$\bg_white \noindent and so on. We call these numbers T_{n} the \emph{triangular numbers} because T_{n} counts how many dots in a triangle with n layers, as drawn above.$

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