Find an explicit rule for the nth term of the sequence. 7, -7, 7, -7

Question

Find an explicit rule for the nth term of the sequence. 7, -7, 7, -7

is it an = 7 • (-1)^n-1

^

Answer 1

Sequence: 7, -7, 7, -7

To find out the explicit rule for the n^th term of the sequence, we have to first realize that there are multiple types of sequences that you need to be familiar with. For example, we would know if it is an "arthmetic sequence" if the sequence if the terms are being added / subtracted by a constant number. On the other hand, we know it is a "geometric sequence" if the terms are being multiplied by a certain number. There are also other types of sequences that exist. In this case, this is a geometric sequence since each term is being multiplied by -1 as we start with 7 as the first number.

There is an explicit formula for the geometric sequence that we can use which is:

a_n = a₁ • r^(n-1)

where a_n is the n^th term, a₁ is the first number in the sequence, r is the ratio (the contant factor that multiplies each term to get the next term in the sequence), and n is the term in the sequence (1^st number means n = 1, 2^nd number means n = 2, etc...).

For the sequence 7, -7, 7, -7, we know that a₁ = 7 since that is the first number. The r can be found by dividing any term by its preceding term. So, we can take the second number and divide by the first number:

r = -7 / 7 = -1

So, the ratio (r) is - 1 and we actually did identify that in the beginning because we realized what the pattern was; it was being multiplied by -1 or it is the same as saying that it is being divided by -1. So, we can just plug in our a₁ and r to get the explicit form for this sequence. Lets write the general geometric sequence explicit form first:

a_n = a₁ • r^(n-1) Now plug in a₁ and r

a_n = 7 • (-1)^(n-1)

Also, we don't need to plug in n since that is the term number so if we are finding a specific term in the sequence; for example, let say you are trying to find the 50^th term in the sequence, then n would be 50 and you would plug that in into the formula to find the 50^th term. So, our final explicit form for this sequence is:

a_n = 7 • (-1)^(n-1)

Answer 2

it's a geometric series with common ratio, r, = -1

nth term is a1(r^(n-1)

= 7(-1)^(n-1)

You're right.