Question

Which function represents this sequence (1,6) (2,18) (3,54) (4,162) (5,486)

Answer 1

The sequence (1,6), (2,18), (3,54), (4,162), (5,486) is represented by a geometric sequence with the function f(n) = 6 × 3^(n-1), where n is the term number.

Step-by-step explanation:

To find which function represents the given sequence (1,6), (2,18), (3,54), (4,162), (5,486), we first observe the relationship between the x-values (the input or independent variable) and the y-values (the output or dependent variable).

Noticing the pattern, each y-value is three times the previous y-value. Mathematically, this is represented by a geometric sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio. In this case, the common ratio is 3.

The nth term (an) of a geometric sequence with the first term a1 and a common ratio r is given by an = a1 × r(n-1). Since the first term (when n=1) is 6, and the common ratio (r) is 3, the nth term of this sequence is given by:

an = 6 × 3(n-1)

This function matches the given sequence. Therefore, the function representing the sequence is:

f(n) = 6 × 3(n-1) (geometric sequence)

Answer 2

The function representing the sequence (1,6) (2,18) (3,54) (4,162) (5,486) is an exponential function given by y = 6 ×
`3^(x-1)`is determined by noticing the pattern that each y-value is three times the y-value of the previous pair.

The student has provided a sequence of ordered pairs and wishes to find the function that represents this sequence: (1,6) (2,18) (3,54) (4,162) (5,486). To determine the pattern, let's look at how each y-value relates to its corresponding x-value. We notice that each y-value is three times the previous y-value, indicating an exponential relationship. Specifically, as we move from one pair to the next, the x-value increases by 1, and the y-value is multiplied by 3, which suggests the formula y = 6 ×
`3^(x-1)`. To validate this, we can plug in the x-values from the sequence into our proposed function:

1. For x = 1: y = 6 ×
   `3^(1-1)` = 6
2. For x = 2: y = 6 ×
   `3^(2-1)` = 18
3. For x = 3: y = 6 ×
   `3^(3-1)`= 54
4. For x = 4: y = 6 ×
   `3^(4-1)` = 162
5. For x = 5: y = 6 ×
   `3^(5-1)` = 486

Since all the pairs satisfy this function, the function that represents the sequence is y = 6 ×
`3^(x-1)`.