Question

Write the two next term:1;3;6;10​

Answer 1

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

Answer 2

Answer:

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

Step-by-step 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]

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

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

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

$\implies8b+2c=5\\\implies4b+c =\frac{5}{2}..........[v]$

[v]-[iv]

$\implies{b=\frac{5}{2}-2=\frac{1}{2}$

Putting this into [iv]

$\frac{3}{2}+c=2\\\implies c=\frac{1}{2}$

Putting everything found so far into [i]

$d=1-b-c=1-\frac{1}{2}-\frac{1}{2}=0$

Hence

$a_n=\frac{1}{2}n^2+\frac{1}{2}n$

$\implies a_n=\frac{1}{2}n(n+1)$

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