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Pth term of an ap is a and qth term is b prove that the sum of its (p q)th?
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Pth term of an ap is a and qth term is b prove that the sum of its (p ...
Let the first term of the AP be 'c' and its common difference be 'd'.
So its pth term is: c + (p - 1)d = a ------ (1)
and its qth term is: c + (q - 1)d = b ----- (2)
Adding (1) & (2): 2c + (p + q - 2)d = a + b
==> 2c + (p + q - 1)d - d = a + b
==> 2c + (p + q - 1)d = (a + b) + d ---------- (3)
Operating (1) - (2): (p - q)d = (a - b)
==> d = (a - b)/(p - q) --------- (4)
Sum to (p + q) terms is:
S₍p+q₎ = {(p + q)/2}*{2c + (p + q - 1)d}
Substituting from (3) and (4), we get
S₍p+q₎ = {(p + q)/2}*{(a + b) + (a - b)/(p - q)} --
hence proved
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Pth term of an ap is a and qth term is b prove that the sum of its (p ...
Proof:
Let's consider an arithmetic progression (AP) with the first term a and the common difference d.
The pth term of this AP can be expressed as:
a + (p-1)d
Similarly, the qth term can be expressed as:
a + (q-1)d
To prove that the sum of the (p+q)th term is equal to (p+q) times the average of the pth and qth term, we need to find the (p+q)th term and the average of the pth and qth term.
Finding the (p+q)th term:
The (p+q)th term can be expressed as:
a + (p+q-1)d
Finding the average of the pth and qth term:
The average of the pth and qth term can be calculated as:
(a + (p-1)d + a + (q-1)d) / 2
Simplifying this expression gives:
(2a + (p+q-2)d) / 2
Proving the equality:
To prove that the sum of the (p+q)th term is equal to (p+q) times the average of the pth and qth term, we need to show that:
(a + (p+q-1)d) = (p+q) * [(2a + (p+q-2)d) / 2]
Multiplying both sides by 2 gives:
2a + 2(p+q-1)d = (p+q) * (2a + (p+q-2)d)
Expanding the right side gives:
2a + 2(p+q-1)d = 2a(p+q) + (p+q)(p+q-2)d
Simplifying both sides gives:
2a + 2pd + 2qd - 2d = 2ap + 2aq + p^2d + pqd + pqd + q^2d - 2pd - 2qd
Cancelling out the terms on both sides gives:
2a - 2d = 2ap + 2aq + p^2d + 2pqd + q^2d - 2pd - 2qd
Further simplifying gives:
2a - 2d = 2ap + 2aq + p^2d + 2pqd + q^2d - 2pd - 2qd
2a - 2d = 2a(p + q) + d(p^2 + 2pq + q^2 -
Let's consider an arithmetic progression (AP) with the first term a and the common difference d.
The pth term of this AP can be expressed as:
a + (p-1)d
Similarly, the qth term can be expressed as:
a + (q-1)d
To prove that the sum of the (p+q)th term is equal to (p+q) times the average of the pth and qth term, we need to find the (p+q)th term and the average of the pth and qth term.
Finding the (p+q)th term:
The (p+q)th term can be expressed as:
a + (p+q-1)d
Finding the average of the pth and qth term:
The average of the pth and qth term can be calculated as:
(a + (p-1)d + a + (q-1)d) / 2
Simplifying this expression gives:
(2a + (p+q-2)d) / 2
Proving the equality:
To prove that the sum of the (p+q)th term is equal to (p+q) times the average of the pth and qth term, we need to show that:
(a + (p+q-1)d) = (p+q) * [(2a + (p+q-2)d) / 2]
Multiplying both sides by 2 gives:
2a + 2(p+q-1)d = (p+q) * (2a + (p+q-2)d)
Expanding the right side gives:
2a + 2(p+q-1)d = 2a(p+q) + (p+q)(p+q-2)d
Simplifying both sides gives:
2a + 2pd + 2qd - 2d = 2ap + 2aq + p^2d + pqd + pqd + q^2d - 2pd - 2qd
Cancelling out the terms on both sides gives:
2a - 2d = 2ap + 2aq + p^2d + 2pqd + q^2d - 2pd - 2qd
Further simplifying gives:
2a - 2d = 2ap + 2aq + p^2d + 2pqd + q^2d - 2pd - 2qd
2a - 2d = 2a(p + q) + d(p^2 + 2pq + q^2 -
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Pth term of an ap is a and qth term is b prove that the sum of its (p q)th?
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Pth term of an ap is a and qth term is b prove that the sum of its (p q)th? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Pth term of an ap is a and qth term is b prove that the sum of its (p q)th? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Pth term of an ap is a and qth term is b prove that the sum of its (p q)th?.
Pth term of an ap is a and qth term is b prove that the sum of its (p q)th? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Pth term of an ap is a and qth term is b prove that the sum of its (p q)th? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Pth term of an ap is a and qth term is b prove that the sum of its (p q)th?.
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