Sure! Let's find the nth term of the sequence 420, -2, -4.
Given the sequence:
- First term (a[tex]\(_1\)[/tex]) = 420
- Second term (a[tex]\(_2\)[/tex]) = -2
- Third term (a[tex]\(_3\)[/tex]) = -4
Let's assume it is an arithmetic sequence. An arithmetic sequence has a constant difference between consecutive terms. This difference is called the common difference ([tex]\(d\)[/tex]).
### Step-by-Step Solution
1. Calculate the common difference (d):
[tex]\[
d = a_2 - a_1
\][/tex]
Given [tex]\(a_1 = 420\)[/tex] and [tex]\(a_2 = -2\)[/tex], we have:
[tex]\[
d = -2 - 420 = -422
\][/tex]
2. Express the nth term formula for an arithmetic sequence:
The nth term ([tex]\(a_n\)[/tex]) of an arithmetic sequence can be calculated using:
[tex]\[
a_n = a_1 + (n - 1)d
\][/tex]
Substituting the values of [tex]\(a_1\)[/tex] and [tex]\(d\)[/tex] into the formula:
[tex]\[
a_n = 420 + (n - 1)(-422)
\][/tex]
3. Simplify the equation:
[tex]\[
a_n = 420 - 422n + 422
\][/tex]
[tex]\[
a_n = 842 - 422n
\][/tex]
So, the nth term of the sequence 420, -2, -4 is given by:
[tex]\[
a_n = 842 - 422n
\][/tex]
### Example:
Let's verify this by calculating the third term ([tex]\(a_3\)[/tex]):
[tex]\[
a_3 = 842 - 422 \times 3
\][/tex]
[tex]\[
a_3 = 842 - 1266
\][/tex]
[tex]\[
a_3 = -424
\][/tex]
