Question

1) Determine if the following sequences are arithmetical, geometric, or neither.
a) 5, 25, 125, 625…
b) 4, 7, 10, 13…
c) 4, 2, 0, -2…

2) For the following arithmetical sequence, determine a1 and d and find an explicit formula for the sequence: -8, -5, -2, 1…
a) What is a1?
b) What is the common difference d?
c) Write an explicit formula for an using the formula for arithmetical sequences an = a1 + (n – 1)d.

Answer 1

(a) 5 x 5 = 25, 25 x 5 = 125, 125 x 5 = 625 -- when you have to multiply to get the next number, it is geometric.

(b) 4 + 3 = 7, 7 + 3 = 10, 10 + 3 = 13 -- when you have to add to get the next number, it is arithmetic

(c) 4 + (-2) = 2, 2 + (-2) = 0, 0 + (-2) = -2 -- added, so its arithmetic

a1 = the first number, so a1 = -8
common difference is 3
I am not sure about the last one...sorry

Answer 2

For the last bit:

\[a_n = a_1+(n-1)d\]You know that \(a_1 = -8,\,d = 3\)
\[a_n = -8 + 3(n-1) = -8 + 3n -3 = 3n-11\]
\[a_n=3n-11\]is your answer. Try it out for various values of \(n\) and make sure you agree it produces your sequence.