Consider the following sequence. __, 1, -4,...

Question

a. Fill in the blank to make the sequence arithmetic.

__, 1, -4...

b. Fill in the blank to make the sequence geometric.

__, 1, -4...

Answer 1

Hi, Jess. Let's do this one.

Let the sequence be: a, 1, -4, b

(a) for an arithmetic sequence the difference, d, between any two consecutive items is constant.

So, d = -4 -1 = -5

Therefore, b = -4 + (-5) =-4 - 5 = -9

And 1 - a = -5 => -a = -6 => a = 6

So, the arithmetic sequence is: 6, 1, -4, -9

(b) for a geometric sequence, the ratio, r, between any two consecutive items is constant.

So, r = -4/1 = -4

Therefore, b/-4 = -4

=> b = 16 (multiplying through by -4)

And 1/a = -4

=> 1 = -4a (multiplying through by a)

=> a = -1/4 (dividing through by -4

So, the geometric sequence is -1/4, 1, -4, 16