2. Which is larger?
The common ratio, r , in a geometric sequence whose second term is
24 and whose fifth term is 1536
or
The common difference, d , in an arithmetic sequence whose fourth
term is 16 and whose seventh term is 31.

The common difference d is larger than the common ratio r

Step-by-step explanation:

Geometric sequence

∵ The second term is 24

∴ = 24

∵

- Equate it by its value

∴ ar = 24 ⇒ (1)

∵ The fifth term is 1536

∴   = 1536

∵

- Equate it by its value

∴ = 1536 ⇒ (2)

Divide (2) by (1)

∴

- Divide up and down by ar

∴ r³ = 64

- Take ∛  for both sides

∴ r = 4

Arithmetic sequence

∵ The fourth term is 16

∴ = 16

∵ = a + (4 - 1)d

∴ = a + 3 d

- Equate it by its value

∴ a + 3d = 16 ⇒ (1)

∵ The seventh term is 31

∴ = 31

∵ = a + (7 - 1)d

∴ = a + 6 d

- Equate it by its value

∴ a + 6 d = 31 ⇒ (2)

Subtract equation (1) from equation (2) to eliminate a and find d

∵ (a - a) + (6 d - 3 d) = (31 - 16)

∴ 3 d = 15

- Divide both sides by 3

∴ d = 5

∵ r = 4 and d = 5

∴ d > r

∴ The common difference d is larger than the common ratio r