How to give a recursive definition and a direct formula and prove that they both are equivalent.

Question

How to give a recursive definition and a direct formula and prove that they both are equivalent.

for example, 10,13,16,19,22,25

I know the formula for this is a,a+d,a+2d,a+3d,...

7,49,343,2401,16807 ,the formula is a,$a^2$,$a^3$,,,,,

but what about recursive definition ?

Answer 1

Define $$a_k=a_0+kn$$ Then, $$a_k-a_{k-1}=a_0+kn-(a_0+(k-1)n)=n$$ $$\iff a_k=n+a_{k-1}$$ Try using the same concept on the geometric series.