The first term of an Arithmetic Progression (AP) is equal to the first term of a Geometric Progression (GP). The second term of the AP is equal to the fourth term of the GP while the tenth term of the AP is equal to the seventh term of the GP .
(a) Given that a is the first term and d is the common difference of the AP while r is the common ratio of the GP, write two equations connecting the AP and the GP.
(b) Find the value of r that satisfies the progressions.
(c) Given that the tenth term of the GP is 5120, find the values of a and d.
(d) Calculate the sum of the first 20 terms of the AP. ​