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Answer :
B is the right answer
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To solve this problem, let's analyze the given arithmetic progression and its properties.
1. The arithmetic progression (A.P) is given as [tex]\(3, 15, 27, \ldots\)[/tex].
To find the common difference [tex]\(d\)[/tex], we can subtract the first term from the second term:
[tex]\[ d = 15 - 3 = 12 \][/tex]
2. The general formula for the [tex]\(n\)[/tex]-th term of an arithmetic progression is:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
where [tex]\(a_1\)[/tex] is the first term, [tex]\(n\)[/tex] is the term number, and [tex]\(d\)[/tex] is the common difference.
For this A.P, the first term [tex]\(a_1 = 3\)[/tex] and the common difference [tex]\(d = 12\)[/tex].
3. We are given that:
[tex]\[ a_p - a_{50} = 180 \][/tex]
Let's calculate [tex]\(a_{50}\)[/tex]:
[tex]\[ a_{50} = a_1 + (50 - 1) \cdot d = 3 + 49 \cdot 12 = 3 + 588 = 591 \][/tex]
4. We need to find the term [tex]\(a_p\)[/tex] such that the difference with [tex]\(a_{50}\)[/tex] is 180:
[tex]\[ a_p - 591 = 180 \][/tex]
[tex]\[ a_p = 180 + 591 = 771 \][/tex]
5. Now, find the term number [tex]\(p\)[/tex] that gives us [tex]\(a_p = 771\)[/tex]:
[tex]\[ a_p = a_1 + (p - 1) \cdot d \][/tex]
[tex]\[ 771 = 3 + (p - 1) \cdot 12 \][/tex]
6. Solving for [tex]\(p\)[/tex]:
[tex]\[ 771 - 3 = (p - 1) \cdot 12 \][/tex]
[tex]\[ 768 = (p - 1) \cdot 12 \][/tex]
[tex]\[ p - 1 = \frac{768}{12} = 64 \][/tex]
[tex]\[ p = 64 + 1 = 65 \][/tex]
Therefore, the value of [tex]\(p\)[/tex] is [tex]\( \boxed{65} \)[/tex].
1. The arithmetic progression (A.P) is given as [tex]\(3, 15, 27, \ldots\)[/tex].
To find the common difference [tex]\(d\)[/tex], we can subtract the first term from the second term:
[tex]\[ d = 15 - 3 = 12 \][/tex]
2. The general formula for the [tex]\(n\)[/tex]-th term of an arithmetic progression is:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
where [tex]\(a_1\)[/tex] is the first term, [tex]\(n\)[/tex] is the term number, and [tex]\(d\)[/tex] is the common difference.
For this A.P, the first term [tex]\(a_1 = 3\)[/tex] and the common difference [tex]\(d = 12\)[/tex].
3. We are given that:
[tex]\[ a_p - a_{50} = 180 \][/tex]
Let's calculate [tex]\(a_{50}\)[/tex]:
[tex]\[ a_{50} = a_1 + (50 - 1) \cdot d = 3 + 49 \cdot 12 = 3 + 588 = 591 \][/tex]
4. We need to find the term [tex]\(a_p\)[/tex] such that the difference with [tex]\(a_{50}\)[/tex] is 180:
[tex]\[ a_p - 591 = 180 \][/tex]
[tex]\[ a_p = 180 + 591 = 771 \][/tex]
5. Now, find the term number [tex]\(p\)[/tex] that gives us [tex]\(a_p = 771\)[/tex]:
[tex]\[ a_p = a_1 + (p - 1) \cdot d \][/tex]
[tex]\[ 771 = 3 + (p - 1) \cdot 12 \][/tex]
6. Solving for [tex]\(p\)[/tex]:
[tex]\[ 771 - 3 = (p - 1) \cdot 12 \][/tex]
[tex]\[ 768 = (p - 1) \cdot 12 \][/tex]
[tex]\[ p - 1 = \frac{768}{12} = 64 \][/tex]
[tex]\[ p = 64 + 1 = 65 \][/tex]
Therefore, the value of [tex]\(p\)[/tex] is [tex]\( \boxed{65} \)[/tex].
