Answer :
To find the first term of the arithmetic sequence, we can use the formula for the n-th term of an arithmetic sequence:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Where:
- [tex]\( a_n \)[/tex] is the n-th term of the sequence.
- [tex]\( a_1 \)[/tex] is the first term.
- [tex]\( d \)[/tex] is the common difference.
- [tex]\( n \)[/tex] is the position of the term in the sequence.
In this problem:
- We know [tex]\( a_{61} = 293 \)[/tex], which is the 61st term.
- The common difference [tex]\( d = -3.5 \)[/tex].
- We need to find [tex]\( a_1 \)[/tex].
Let's rearrange the formula to solve for [tex]\( a_1 \)[/tex]:
[tex]\[ a_1 = a_n - (n - 1) \cdot d \][/tex]
Substitute the known values into the formula:
[tex]\[ a_1 = 293 - (61 - 1) \cdot (-3.5) \][/tex]
Calculate the expression inside the parentheses:
[tex]\[ a_1 = 293 - 60 \cdot (-3.5) \][/tex]
Now, calculate [tex]\( 60 \cdot (-3.5) \)[/tex]:
[tex]\[ 60 \cdot (-3.5) = -210 \][/tex]
Substitute back into the equation:
[tex]\[ a_1 = 293 - (-210) \][/tex]
This becomes:
[tex]\[ a_1 = 293 + 210 \][/tex]
Finally, add them together:
[tex]\[ a_1 = 503 \][/tex]
Thus, the first term of the arithmetic sequence is [tex]\( 503 \)[/tex].
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Where:
- [tex]\( a_n \)[/tex] is the n-th term of the sequence.
- [tex]\( a_1 \)[/tex] is the first term.
- [tex]\( d \)[/tex] is the common difference.
- [tex]\( n \)[/tex] is the position of the term in the sequence.
In this problem:
- We know [tex]\( a_{61} = 293 \)[/tex], which is the 61st term.
- The common difference [tex]\( d = -3.5 \)[/tex].
- We need to find [tex]\( a_1 \)[/tex].
Let's rearrange the formula to solve for [tex]\( a_1 \)[/tex]:
[tex]\[ a_1 = a_n - (n - 1) \cdot d \][/tex]
Substitute the known values into the formula:
[tex]\[ a_1 = 293 - (61 - 1) \cdot (-3.5) \][/tex]
Calculate the expression inside the parentheses:
[tex]\[ a_1 = 293 - 60 \cdot (-3.5) \][/tex]
Now, calculate [tex]\( 60 \cdot (-3.5) \)[/tex]:
[tex]\[ 60 \cdot (-3.5) = -210 \][/tex]
Substitute back into the equation:
[tex]\[ a_1 = 293 - (-210) \][/tex]
This becomes:
[tex]\[ a_1 = 293 + 210 \][/tex]
Finally, add them together:
[tex]\[ a_1 = 503 \][/tex]
Thus, the first term of the arithmetic sequence is [tex]\( 503 \)[/tex].
