Question

Arithmetic Sequences

What is the explicit formula for the sequence 5, 7, 9, 11, 13…?
a) an = 5n
b) an = 2n
c) an = 3n + 2
d) an = 5n + 2

What do we know about this sequence?
a) It's an arithmetic sequence with a common difference of 1.
b) It's an arithmetic sequence with a common difference of 2.
c) It's a geometric sequence with a common ratio of 2.
d) It's a geometric sequence with a common ratio of 1.

First find the common difference: d = 2
What is the first term? a1 = 5

Now we'll plug those into our general rule for arithmetic sequences:
an = a1 + (n - 1)d
an = 5 + 2(n - 1)

Now distribute and simplify.

Explicit form is:
a) an = 5 + 2n - 2
b) an = 3 + 2n
c) an = 5n - 2
d) an = 2n + 3

Answer 1

The explicit formula for the arithmetic sequence 5, 7, 9, 11, 13... is an = 2n + 3. The sequence has a common difference of 2.

Step-by-step explanation:

The sequence given is 5, 7, 9, 11, 13..., which is an example of an arithmetic sequence.

The explicit formula for this sequence can be found by identifying the first term and the common difference. The first term (a1) is 5, and since each subsequent term increases by 2, the common difference (d) is 2. The explicit formula for an arithmetic sequence is given by an = a1 + (n - 1)d. By plugging in the values of a1 and d, we get:

an = 5 + (n - 1) * 2

Distributing the 2, we simplify the expression to:

an = 5 + 2n - 2

Further simplification gives us the final explicit formula:

an = 2n + 3

So the correct answer is indeed an = 2n + 3.

We also confirm that this sequence is an arithmetic sequence with a common difference of 2, therefore the second part of the question can be answered as choice (b): It's an arithmetic sequence with a common difference of 2.