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Answer :
Let's solve the given problem step-by-step.
We are given an arithmetic progression (AP) where the terms are:
[tex]\[ -3, -7, -11, \ldots \][/tex]
First, let's identify the first term and the common difference of this AP.
- The first term, [tex]\( a_1 \)[/tex], is [tex]\(-3\)[/tex].
- The common difference, [tex]\( d \)[/tex], can be found by subtracting the first term from the second term:
[tex]\[ d = -7 - (-3) = -7 + 3 = -4 \][/tex]
We need to find [tex]\( a_{30} - a_{20} \)[/tex] directly without actually finding [tex]\( a_{20} \)[/tex] and [tex]\( a_{30} \)[/tex].
Step-by-Step Solution:
1. We know the formula for the [tex]\( n \)[/tex]-th term of an arithmetic progression:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
2. Therefore, the 30th term ([tex]\( a_{30} \)[/tex]) can be written as:
[tex]\[ a_{30} = a_1 + 29 \cdot d \][/tex]
Similarly, the 20th term ([tex]\( a_{20} \)[/tex]) can be written as:
[tex]\[ a_{20} = a_1 + 19 \cdot d \][/tex]
3. To find [tex]\( a_{30} - a_{20} \)[/tex], we subtract [tex]\( a_{20} \)[/tex] from [tex]\( a_{30} \)[/tex]:
[tex]\[ a_{30} - a_{20} = (a_1 + 29 \cdot d) - (a_1 + 19 \cdot d) \][/tex]
4. Simplifying this expression:
[tex]\[ a_{30} - a_{20} = a_1 + 29 \cdot d - a_1 - 19 \cdot d \][/tex]
[tex]\[ a_{30} - a_{20} = 29 \cdot d - 19 \cdot d \][/tex]
[tex]\[ a_{30} - a_{20} = (29 - 19) \cdot d \][/tex]
[tex]\[ a_{30} - a_{20} = 10 \cdot d \][/tex]
5. Substituting the value of [tex]\( d \)[/tex] which is [tex]\(-4\)[/tex]:
[tex]\[ a_{30} - a_{20} = 10 \cdot (-4) \][/tex]
[tex]\[ a_{30} - a_{20} = -40 \][/tex]
Hence, the difference between the 30th term and the 20th term ([tex]\( a_{30} - a_{20} \)[/tex]) is [tex]\(-40\)[/tex].
We are given an arithmetic progression (AP) where the terms are:
[tex]\[ -3, -7, -11, \ldots \][/tex]
First, let's identify the first term and the common difference of this AP.
- The first term, [tex]\( a_1 \)[/tex], is [tex]\(-3\)[/tex].
- The common difference, [tex]\( d \)[/tex], can be found by subtracting the first term from the second term:
[tex]\[ d = -7 - (-3) = -7 + 3 = -4 \][/tex]
We need to find [tex]\( a_{30} - a_{20} \)[/tex] directly without actually finding [tex]\( a_{20} \)[/tex] and [tex]\( a_{30} \)[/tex].
Step-by-Step Solution:
1. We know the formula for the [tex]\( n \)[/tex]-th term of an arithmetic progression:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
2. Therefore, the 30th term ([tex]\( a_{30} \)[/tex]) can be written as:
[tex]\[ a_{30} = a_1 + 29 \cdot d \][/tex]
Similarly, the 20th term ([tex]\( a_{20} \)[/tex]) can be written as:
[tex]\[ a_{20} = a_1 + 19 \cdot d \][/tex]
3. To find [tex]\( a_{30} - a_{20} \)[/tex], we subtract [tex]\( a_{20} \)[/tex] from [tex]\( a_{30} \)[/tex]:
[tex]\[ a_{30} - a_{20} = (a_1 + 29 \cdot d) - (a_1 + 19 \cdot d) \][/tex]
4. Simplifying this expression:
[tex]\[ a_{30} - a_{20} = a_1 + 29 \cdot d - a_1 - 19 \cdot d \][/tex]
[tex]\[ a_{30} - a_{20} = 29 \cdot d - 19 \cdot d \][/tex]
[tex]\[ a_{30} - a_{20} = (29 - 19) \cdot d \][/tex]
[tex]\[ a_{30} - a_{20} = 10 \cdot d \][/tex]
5. Substituting the value of [tex]\( d \)[/tex] which is [tex]\(-4\)[/tex]:
[tex]\[ a_{30} - a_{20} = 10 \cdot (-4) \][/tex]
[tex]\[ a_{30} - a_{20} = -40 \][/tex]
Hence, the difference between the 30th term and the 20th term ([tex]\( a_{30} - a_{20} \)[/tex]) is [tex]\(-40\)[/tex].
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