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

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.
