Determine which arrangement will have 703 dots In a quadratic sequence of arrangement 1 with 1 dot, arrangement 2 with 3 dots , arrangement 3 with 6 dots and arrangement 4 with 10 dots

Question

Quadratic sequence

Answer 1

Assume sequence is given by q(n)=an²+bn+c.

q(1)=1=a+b+c; q(2)=3=4a+2b+c; q(3)=6=9a+3b+c; q(4)=10=16a+4b+c.

q(2)-q(1)=2=3a+b; q(3)-q(1)=5=8a+2b.

So 4=6a+2b and 5=8a+2b, therefore 5-4=1=8a-6a=2a, a=½.

3a+b=2, 3/2+b=2, b=2-3/2=½.

a+b+c=1, ½+½+c=1, making c=0 and q(n)=n²/2+n/2=½n(n+1).

Let's check out q(4)=4×5/2=10 which matches what was given, confirming the quadratic sequence formula q(n)=½n(n+1).

For example, q(5)=15 from the formula. This is the formula for the sum of the natural numbers up to n:

1, 1+2=3, 1+2+3=6, 1+2+3+4=10, 1+2+3+4+5=15, etc.