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,
[tex]a_n=bn^2+cn+d[/tex]
Putting [tex]n=1,2,3...[/tex]
[tex]b+c+d=1....[i][/tex]
[tex]4b+2c+d=3......[ii][/tex]
[tex]9b+3c+d=6....[iii][/tex]
From equation [i]
[tex]d=1-b-c[/tex]
Putting this into [ii] and [iii]
[tex]4b+2c+1-b-c=3[/tex]
[tex]\implies3b+c=2....[iv][/tex]
[tex]9b+3c+1-b-c=6[/tex]
[tex]\implies8b+2c=5\\\implies4b+c =\frac{5}{2}..........[v][/tex]
[v]-[iv]
[tex]\implies{b=\frac{5}{2}-2=\frac{1}{2}[/tex]
Putting this into [iv]
[tex]\frac{3}{2}+c=2\\\implies c=\frac{1}{2}[/tex]
Putting everything found so far into [i]
[tex]d=1-b-c=1-\frac{1}{2}-\frac{1}{2}=0\\[/tex]
Hence
[tex]a_n=\frac{1}{2}n^2+\frac{1}{2}n[/tex]
[tex]\implies a_n=\frac{1}{2}n(n+1)[/tex]
and this matches with the general term of the triangular number sequence.
