Question

Which function forms a geometric sequence when x = 1, 2, 3, . . . ?

Answer 1

Answer:

The function  $f(x)=3(4)^{x}$ forms a geometric sequence ⇒ 1st answer

Step-by-step explanation:

In the geometric sequence there is a common ratio between each two consecutive terms

Lets substitute x by 1, 2, 3 to find the first three terms and check if there is a common ratio between the consecutive terms or not

∵ $f(x)=3(4)^{x}$

∵ x = 1, 2 , 3

- Substitute x by 1

∴  $f(1)=3(4)^{1}=3(4)=12$

- Substitute x by 2

∴  $f(2)=3(4)^{2}=3(16)=48$

- Substitute x by 3

∴  $f(3)=3(4)^{3}=3(64)=192$

∴ The sequence is 12, 48, 192, .......

- Let us check the ratio between each two consecutive terms

∵ 48 ÷ 12 = 4

∵ 192 ÷ 48 = 4

- There is a constant ratio 4 between the consecutive terms

∴ The function  $f(x)=3(4)^{x}$ forms a geometric sequence

∵ f(x) = 3(x)²

∵ x = 1, 2 , 3

- Substitute x by 1

∴ f(1) = 3(1)² = 3(1) = 3

- Substitute x by 2

∴ f(2) = 3(2)² = 3(4) = 12

- Substitute x by 3

∴ f(3) = 3(3)² = 3(9) = 27

∴ The sequence is 3, 12, 27, .......

- Let us check the ratio between each two consecutive terms

∵ 12 ÷ 3 = 4

∵ 27 ÷ 12 = 2.25

- There is no constant ratio 4 between the consecutive terms

∴ The function f(x) = 3(x)² does not form a geometric sequence

∵ f(x) = 2x + 4

∵ x = 1, 2, 3

- Substitute x by 1

∴ f(1) = 2(1) + 4 = 2 + 4 = 6

- Substitute x by 2

∴ f(2) = 2(2) + 4 = 4 + 4 = 8

- Substitute x by 3

∴ f(3) = 2(3) + 4 = 6 + 4 = 10

∴ The sequence is 6, 8, 10, .......

- Let us check the ratio between each two consecutive terms

∵ 8 ÷ 6 = $\frac{4}{3}$

∵ 10 ÷ 8 = $\frac{5}{4}$

- There is no constant ratio 4 between the consecutive terms

∴ The function f(x) = 2x + 4 does not form a geometric sequence

∵ f(x) = x + $2^{4}$

∵ x = 1, 2, 3

- Substitute x by 1

∴ f(1) = 1 + $2^{4}$  = 1 + 16 = 17

- Substitute x by 2

∴ f(2) = 2 + $2^{4}$  = 2 + 16 = 18

- Substitute x by 3

∴ f(3) = 3 + $2^{4}$  = 3 + 16 = 19

∴ The sequence is 17, 18, 19, .......

- There is a common difference 1 (not a common ratio) between

   the consecutive terms

∴ The function f(x) = x + $2^{4}$ does not form a geometric sequence

Answer 2

the answer is A

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