The following sequence of perfect squares is not arithmetic.
1, 4, 9, 16, 25, 36, 49, 64, 81, . . .
The related sequence formed from the first differences of this sequence, however, is arithmetic.
(a) Write the first eight terms of the related arithmetic sequence described above. What is the nth term of this sequence?
(b) Explain how to find an arithmetic sequence that is related to the following sequence of perfect cubes. 1, 8, 27, 64, 125, 216, 343, 512, 729, . . .
(c) Write the first seven terms of the related arithmetic sequence in part (b) and find the nth term of the sequence.
(d) Explain how to find an arithmetic sequence that is related to the following sequence of perfect fourth powers. 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, . . .
(e) Write the first six terms of the related arithmetic sequence in part (d) and find the nth term of the sequence.