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27. Write the following arithmetic sequence using an explicit formula:

[tex]\[ a_1 = 8, \quad a_n = a_{n-1} - 2 \][/tex]

A. [tex]\[ a_n = 2 - 8(n-1) \][/tex]
B. [tex]\[ a_n = 8 - 2(n-1) \][/tex]
C. [tex]\[ a_n = 2 + 8(n-1) \][/tex]
D. [tex]\[ a_n = 8 + 2(n-1) \][/tex]

Sagot :

GinnyAnsweridn
Let's derive the explicit formula for the given arithmetic sequence step-by-step:

1. We are given that the first term [tex]\( a_1 \)[/tex] of the sequence is 8:
[tex]\[ a_1 = 8 \][/tex]

2. The sequence is defined recursively with the relationship:
[tex]\[ a_n = a_{n-1} - 2 \][/tex]
This indicates that the common difference [tex]\( d \)[/tex] is -2.

3. To find the explicit formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence, we use the general formula:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]

4. Substituting the known values ([tex]\( a_1 = 8 \)[/tex] and [tex]\( d = -2 \)[/tex]) into the formula:
[tex]\[ a_n = 8 + (n - 1) \cdot (-2) \][/tex]

5. Simplify the expression inside the parentheses:
[tex]\[ a_n = 8 - 2(n - 1) \][/tex]

Therefore, the explicit formula for the given arithmetic sequence is:
[tex]\[ a_n = 8 - 2(n - 1) \][/tex]

Among the provided choices, the correct formula is:
[tex]\[ a_n = 8 - 2(n - 1) \][/tex]

So, the correct answer is:
[tex]\[ a_n = 8-2(n-1) \][/tex]
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