An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. To find the explicit formula for the arithmetic sequence given, we need to determine the first term and the common difference.
In the sequence 75, 9, 105, 12, let’s identify the terms:
- First term (a1) = 75
- Second term (a2) = 9
- Third term (a3) = 105
- Fourth term (a4) = 12
To find the common difference, we subtract the first term from the second term:
d = a2 - a1 = 9 - 75 = -66Then, let’s check the difference between other pairs of terms:
d = a3 - a2 = 105 - 9 = 96d = a4 - a3 = 12 - 105 = -93It appears that there is no constant common difference here. This indicates that the terms provided do not form a proper arithmetic sequence, as the differences between them are not consistent. However, if we were hypothetically proceeding with the first term and assuming some constant difference, the explicit formula for an arithmetic series would be:
an = a1 + (n - 1) * dPlugging the values would yield:
an = 75 + (n - 1) * dIn conclusion, since the sequence does not meet the criteria for an arithmetic sequence, it cannot have a valid explicit formula. If you meant a different set of numbers or if the intention was to provide a different sequence, please clarify, and we can examine it further.