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The sum of (p + q)th and (p – q)th terms of an AP is equal to
- a)(2p)th term
- b)(2q)th term
- c)Twice the pth term
- d)Twice the qth term
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
The sum of (p + q)th and (p – q)th terms of an AP is equal toa)(...
Concept:
The nth term of an AP is given by: Tn = a + (n - 1) × d, where a = first term and d = common difference.
The nth term of an AP is given by: Tn = a + (n - 1) × d, where a = first term and d = common difference.
Calculation:
As we know that, the nth term of an AP is given by: Tn = a + (n - 1) × d, where a = first term and d = common difference.
Let a be the first term and d is the common difference.
As we know that, the nth term of an AP is given by: Tn = a + (n - 1) × d, where a = first term and d = common difference.
Let a be the first term and d is the common difference.
⇒ ap+q = a+(p + q − 1) × d ...1)
⇒ ap−q = a + (p − q − 1) × d ...2)
By adding (1) and (2), we get
⇒ ap+q + ap−q = 2a + 2(p−1)d = 2 × [a + (p−1)d] = 2 × ap
Most Upvoted Answer
The sum of (p + q)th and (p – q)th terms of an AP is equal toa)(...
Understanding the Problem
In an arithmetic progression (AP), the nth term can be expressed as:
- Tn = a + (n - 1)d, where 'a' is the first term and 'd' is the common difference.
We need to analyze the sum of the (p + q)th and (p - q)th terms of this AP.
Calculating the Terms
- The (p + q)th term:
- T(p + q) = a + (p + q - 1)d
- The (p - q)th term:
- T(p - q) = a + (p - q - 1)d
Finding the Sum
- The sum of the (p + q)th and (p - q)th terms:
Sum = T(p + q) + T(p - q)
= [a + (p + q - 1)d] + [a + (p - q - 1)d]
= 2a + (2p - 2)d
= 2[a + (p - 1)d]
= 2T(p)
Conclusion
The sum of the (p + q)th and (p - q)th terms equals twice the pth term of the AP. Hence, the correct answer is option 'C': "Twice the pth term."
This relationship shows how specific terms in an AP can interact, reinforcing the properties of arithmetic sequences.
In an arithmetic progression (AP), the nth term can be expressed as:
- Tn = a + (n - 1)d, where 'a' is the first term and 'd' is the common difference.
We need to analyze the sum of the (p + q)th and (p - q)th terms of this AP.
Calculating the Terms
- The (p + q)th term:
- T(p + q) = a + (p + q - 1)d
- The (p - q)th term:
- T(p - q) = a + (p - q - 1)d
Finding the Sum
- The sum of the (p + q)th and (p - q)th terms:
Sum = T(p + q) + T(p - q)
= [a + (p + q - 1)d] + [a + (p - q - 1)d]
= 2a + (2p - 2)d
= 2[a + (p - 1)d]
= 2T(p)
Conclusion
The sum of the (p + q)th and (p - q)th terms equals twice the pth term of the AP. Hence, the correct answer is option 'C': "Twice the pth term."
This relationship shows how specific terms in an AP can interact, reinforcing the properties of arithmetic sequences.
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The sum of (p + q)th and (p – q)th terms of an AP is equal toa)(2p)th termb)(2q)th termc)Twice the pth termd)Twice the qth termCorrect answer is option 'C'. Can you explain this answer?
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