Answer :
To find the 72nd term of the arithmetic sequence given by the terms 11, 7, 3,..., we follow these steps:
1. Identify the first term (a₁):
The first term of the sequence is \( a₁ = 11 \).
2. Determine the common difference (d):
The common difference \( d \) can be found by subtracting the first term from the second term:
[tex]\[ d = 7 - 11 = -4 \][/tex]
3. Use the formula for the nth term of an arithmetic sequence:
The nth term \( a_n \) of an arithmetic sequence can be calculated using the formula:
[tex]\[ a_n = a₁ + (n-1) \cdot d \][/tex]
Here, \( n = 72 \). Therefore, we need to substitute \( n = 72 \), \( a₁ = 11 \), and \( d = -4 \) into the formula.
4. Substitute the values and solve:
[tex]\[ a_{72} = 11 + (72 - 1) \cdot (-4) \][/tex]
Simplify within the parentheses:
[tex]\[ a_{72} = 11 + (71) \cdot (-4) \][/tex]
Multiply 71 by -4:
[tex]\[ a_{72} = 11 + (-284) \][/tex]
Add 11 to -284:
[tex]\[ a_{72} = 11 - 284 \][/tex]
This results in:
[tex]\[ a_{72} = -273 \][/tex]
So, the 72nd term of the arithmetic sequence is [tex]\(-273\)[/tex].
1. Identify the first term (a₁):
The first term of the sequence is \( a₁ = 11 \).
2. Determine the common difference (d):
The common difference \( d \) can be found by subtracting the first term from the second term:
[tex]\[ d = 7 - 11 = -4 \][/tex]
3. Use the formula for the nth term of an arithmetic sequence:
The nth term \( a_n \) of an arithmetic sequence can be calculated using the formula:
[tex]\[ a_n = a₁ + (n-1) \cdot d \][/tex]
Here, \( n = 72 \). Therefore, we need to substitute \( n = 72 \), \( a₁ = 11 \), and \( d = -4 \) into the formula.
4. Substitute the values and solve:
[tex]\[ a_{72} = 11 + (72 - 1) \cdot (-4) \][/tex]
Simplify within the parentheses:
[tex]\[ a_{72} = 11 + (71) \cdot (-4) \][/tex]
Multiply 71 by -4:
[tex]\[ a_{72} = 11 + (-284) \][/tex]
Add 11 to -284:
[tex]\[ a_{72} = 11 - 284 \][/tex]
This results in:
[tex]\[ a_{72} = -273 \][/tex]
So, the 72nd term of the arithmetic sequence is [tex]\(-273\)[/tex].
