Question

You having a discussion about sequences with your classmate she insists that the sequence two and three and five and eight and 12 must be either arithmetic or geometric is she correct or incorrect explain please(I really need help)

Answer 1

2,3,5,8,12

dy=1,2,3,4 there is no common difference so this is not an arithmetic sequence.

2/1, 5/3, 8/5, 12/8 there is no common ratio so this is not a geometric sequence...

however,

d2y=1,1,1 which means that there is a constant acceleration, the sequence then is the result of a quadratic equation....

So we only need three equations to solve for a,b, and c of ax^2+bx+c=y

9a+3b+c=5, 4a+2b+c=3, a+b+c=2 getting differences...

5a+b=2, 3a+b=1 and again...

2a=1, so a=1/2 making 5a+b=2 become:

5/2+b=2, so b=-1/2 making a+b+c=2 become:

1/2-1/2+c=2, so c=2, thus the formula for the sequence is:

a(n)=(n^2-n+4)/2

So it is quadratic. It is not arithmetic or geometric.