The pressure is so overwhelming please help me immediately I need help with my math homework

Question

Identify a possible explicit rule for the nth term of the sequence 5, 3, 1, −1, −3, ….

A.) a_n = 2n + 3

B.) a_n = n + 4

C.) a_n = 7 − 2n

D.) a_n = 2n + 1

please solve and explain

Answer 1

Heya.

So a sequence is just a bunch of numbers that are changing in some order.

The nth term is just a way of saying the 1st term, 2nd term, 3rd term, etc. where n = 1, 2, 3, etc.

a_n represents the value at the nth term (1st term, 2nd term, etc.)

In our sequence here: 5, 3, 1, ...

The 1st term (at n = 1) is a₁, which is 5

The 2nd term (at n = 2) is a₂, which is 3

etc.

So the first thing you want to do in a problem with sequences is look for a pattern.

Here we see that each term decreases by 2.

5

5 - 2 = 3

3 - 2 = 1

1 - 2 = -1

etc.

When each term in a sequence goes down or up by the same amount, it means that each term can be represented by n multiplied by that amount.

So as a simple example to demonstrate this, if a sequence was a_n = 2n

The 1st term (n=1) would be a₁ = 2(1) = 2

The 2nd term (n=2) would be a₂ = 2(2) = 4

The 3rd term (n=3) would be a₃ = 2(3) = 6

In the sequence 2, 4, 6, ... we can see that each term increases by 2.

Since in our sequence, each term decreases by 2, we can represent this as -2n.

Given that the question is multiple choice, only one of the options has -2n, choice C, and therefore must be the correct answer. However, we will continue explaining how to get the answer completely.

So we know that -2n must be in the equation, but if the sequence were simply a_n = -2n, when we plug in at n = 1 (the first term), we get a₁ = -2. And -2 definitely does not equal 5.

So we know that the first term must have some constant added to it, in order to get to 5.

We express that as:

a_n = -2n + x, where x is the constant that is added to the term.

Now we can work backwards, from the fact that the first term is 5.

a₁ = 5

a₁ = -2(1) + x | I'm just replacing the formula above with n = 1 for the first term

5 = -2(1) + x | Now we just solve for x

5 = -2 + x

7 = x

Great! Now we know our constant that is added is 7.

We can put that back into the original equation:

a_n = -2n + 7

a_n = 7 - 2n

If we want to double check, we can check the 2nd and 3rd terms.

2nd term: a₂ = 7 - 2(2) = 7 - 4 = 3

3rd term: a₃ = 7 - 2(3) = 7 - 6 = 1