Answer :
To find the first term [tex]\( a_1 \)[/tex] of an arithmetic sequence where [tex]\( a_{31} = 197 \)[/tex] and the common difference [tex]\( d = 10 \)[/tex], we can use the formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
In this case, we are given:
- [tex]\( a_{31} = 197 \)[/tex]
- [tex]\( n = 31 \)[/tex]
- [tex]\( d = 10 \)[/tex]
We need to find [tex]\( a_1 \)[/tex].
Substitute the given values into the formula:
[tex]\[ a_{31} = a_1 + (31 - 1) \cdot 10 \][/tex]
[tex]\[ 197 = a_1 + 30 \cdot 10 \][/tex]
[tex]\[ 197 = a_1 + 300 \][/tex]
To solve for [tex]\( a_1 \)[/tex], subtract 300 from both sides of the equation:
[tex]\[ 197 - 300 = a_1 \][/tex]
[tex]\[ -103 = a_1 \][/tex]
So, the first term [tex]\( a_1 \)[/tex] of the sequence is [tex]\( -103 \)[/tex].
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
In this case, we are given:
- [tex]\( a_{31} = 197 \)[/tex]
- [tex]\( n = 31 \)[/tex]
- [tex]\( d = 10 \)[/tex]
We need to find [tex]\( a_1 \)[/tex].
Substitute the given values into the formula:
[tex]\[ a_{31} = a_1 + (31 - 1) \cdot 10 \][/tex]
[tex]\[ 197 = a_1 + 30 \cdot 10 \][/tex]
[tex]\[ 197 = a_1 + 300 \][/tex]
To solve for [tex]\( a_1 \)[/tex], subtract 300 from both sides of the equation:
[tex]\[ 197 - 300 = a_1 \][/tex]
[tex]\[ -103 = a_1 \][/tex]
So, the first term [tex]\( a_1 \)[/tex] of the sequence is [tex]\( -103 \)[/tex].
