Question

Write the two next term:1;3;6;10​

Answer 1

1, 3, 6, 10, 15, 21....

Explanation:

What is the formula of the sequence

Since the difference in the second row matches up, the nth term of the sequence has the general form of a second degree polynomial.

So,

`a_n=bn^2+cn+d`

Putting
`n=1,2,3...`

`b+c+d=1....[i]`

`4b+2c+d=3......[ii]`

`9b+3c+d=6....[iii]`

From equation [i]

`d=1-b-c`

Putting this into [ii] and [iii]

`4b+2c+1-b-c=3`

`\implies3b+c=2....[iv]`

`9b+3c+1-b-c=6`

`\implies8b+2c=5\\\implies4b+c =(5)/(2)..........[v]`

[v]-[iv]

`\implies{b=(5)/(2)-2=(1)/(2)`

Putting this into [iv]

`(3)/(2)+c=2\\\implies c=(1)/(2)`

Putting everything found so far into [i]

`d=1-b-c=1-(1)/(2)-(1)/(2)=0\\`

Hence

`a_n=(1)/(2)n^2+(1)/(2)n`

`\implies a_n=(1)/(2)n(n+1)`

and this matches with the general term of the triangular number sequence.

Answer 2

15 and 21. From 1 to 3 is plus 2 and from 3 to 6 is plus 3 and from 6 to 10 is plus 4 and so on