Arithmetic Sequence

Question

The terms of sequence called the triangle numbers are the sums of the consecutive positive integers. The first term of the sequence is defined as 1. The second term is the sum of 1+2=3, the third term is 1+2+3=6, and so on. The first 4 terms of the sequence are 

 

1,3,6,10...

 

What is the 100th triangular number?

 

 

Answer 1

 

look at the following...

 

1   2   3   4   5   6   7

7   6   5   4   3   2   1

add the columns

8   8   8   8   8   8   8

there are 7 8's, so 7*8=56 which is two times the sum of 1  2  3  4  5  6  7

1+2+3+4+5+6+7=28 and 56/2=28

1

1+2=3

1+2+3=6

1+2+3+4=10

and so on

 

1       2       3       4       5..... n

n     n-1    n-2     n-3    n-4...1

add the columns

n+1  n+1   n+1   n+1   n+1.....n+1

there are n of the (n+1)'s

but n*(n+1) is twice the sum of the numbers from 1 to n so you divide the product by 2

n(n+1)/2 is the formula you use

the triangular numbers are:

1(1+1)/2=1

2(2+1)/2=3

3(3+1)/2=6

4(4+1)/2=10

5(5+1)/2=15

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100(100+1)/2=5050 is the 100th triangular number