Question

Tony is working on an arithmetic sequence where t(1) =4 and t(8) =67 Write the explicit equation and recursive equation for this sequence. Is 500 a term of the sequence? Explain why or why not.

Answer 1

Assuming t(1) is our first term, we can say:

t(2)=t(1)+d

t(3)=t(2)+d

t(4)=t(3)+d

This can go on forever, we can then establish the recursive formula which states:

t(n+1)=t(n)+d

or

t(n)=t(n-1)+d

Explicit formula:

t(2)=t(1)+d

t(3)=t(2)+d, but t(2)=t(1)+d

t(3)=t(1)+d+d

t(3)=t(1)+2d

t(4)=t(1)+3d

t(n)=t(1)+(n-1)d

We can calculate d for these equations:

t(8)=t(1)+7d

67=4+7d

7d=63

d=9

Explicit relation:

t(n)=4+9(n-1)

t(n)=9n-5

Recursive relation:

t(n)=t(n-1)+9

t(500)=9(500)-5=4500-5=4495