To determine which of the given sequences is an arithmetic sequence with the first term [tex]\( a_1 = 14 \)[/tex] and the common difference [tex]\( d = -3 \)[/tex], follow these steps:
1. An arithmetic sequence is defined by the property that the difference between consecutive terms is constant and equal to the common difference [tex]\( d \)[/tex].
2. Verify each sequence by checking if the difference between consecutive terms is [tex]\( -3 \)[/tex].
Sequence 1: [tex]\( 14, 11, 8, 5, \ldots \)[/tex]
- First term: [tex]\( a_1 = 14 \)[/tex]
- Second term: [tex]\( a_2 = 11 \)[/tex]
[tex]\[ a_2 - a_1 = 11 - 14 = -3 \][/tex]
- Third term: [tex]\( a_3 = 8 \)[/tex]
[tex]\[ a_3 - a_2 = 8 - 11 = -3 \][/tex]
- Fourth term: [tex]\( a_4 = 5 \)[/tex]
[tex]\[ a_4 - a_3 = 5 - 8 = -3 \][/tex]
Since the difference between consecutive terms is consistently [tex]\(-3\)[/tex], Sequence 1 is indeed an arithmetic sequence with [tex]\( a_1 = 14 \)[/tex] and [tex]\( d = -3 \)[/tex].
Sequence 2: [tex]\( 14, 17, 20, 23, \ldots \)[/tex]
- First term: [tex]\( a_1 = 14 \)[/tex]
- Second term: [tex]\( a_2 = 17 \)[/tex]
[tex]\[ a_2 - a_1 = 17 - 14 = 3 \][/tex]
The difference here is [tex]\( 3 \)[/tex], not [tex]\(-3\)[/tex]. Therefore, Sequence 2 is not an arithmetic sequence with [tex]\( a_1 = 14 \)[/tex] and [tex]\( d = -3 \)[/tex].
Sequence 3: [tex]\( 3, 17, 31, 45, \ldots \)[/tex]
- First term: [tex]\( a_1 = 3 \)[/tex]
The first term is not [tex]\( 14 \)[/tex], and subsequent differences are not [tex]\(-3\)[/tex]. Therefore, Sequence 3 is not an arithmetic sequence with [tex]\( a_1 = 14 \)[/tex] and [tex]\( d = -3 \)[/tex].
Sequence 4: [tex]\( -3, 11, 25, 39, \ldots \)[/tex]
- First term: [tex]\( a_1 = -3 \)[/tex]
The first term is not [tex]\( 14 \)[/tex], and subsequent differences are not [tex]\(-3\)[/tex]. Therefore, Sequence 4 is not an arithmetic sequence with [tex]\( a_1 = 14 \)[/tex] and [tex]\( d = -3 \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
