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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 pthterm
- d)Twice the qthterm
Correct answer is option 'C'. Can you explain this answer?
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To understand why the correct answer is option 'C', let's break down the problem step by step.
Let's assume that the common difference of the arithmetic progression (AP) is 'd'. The 'p' and 'q' terms of the AP can be represented as:
(p)th term = a + (p-1)d
(q)th term = a + (q-1)d
where 'a' is the first term of the AP.
According to the given condition, the sum of the (p+q)th term and (p-q)th term is equal to the (2p)th term:
(a + (p+q-1)d) + (a + (p-q-1)d) = a + (2p-1)d
Simplifying the equation:
2a + 2pd + 2qd - 2d = a + 2pd - d
Cancelling out the common terms:
2qd - 2d = -d
Rearranging the terms:
2qd = d
Dividing both sides of the equation by 'd':
2q = 1
Simplifying further:
q = 1/2
Now, let's find the (p/2)th term of the AP using the formula:
(p/2)th term = a + ((p/2)-1)d
Substituting the value of 'q' into the equation:
(p/2)th term = a + ((1/2)-1)d
= a - (1/2)d
Comparing this with the (p)th term of the AP:
(p)th term = a + (p-1)d
We can see that the (p/2)th term is half of the (p)th term. Therefore, the correct answer is option 'C': Twice the pth term.
Let's assume that the common difference of the arithmetic progression (AP) is 'd'. The 'p' and 'q' terms of the AP can be represented as:
(p)th term = a + (p-1)d
(q)th term = a + (q-1)d
where 'a' is the first term of the AP.
According to the given condition, the sum of the (p+q)th term and (p-q)th term is equal to the (2p)th term:
(a + (p+q-1)d) + (a + (p-q-1)d) = a + (2p-1)d
Simplifying the equation:
2a + 2pd + 2qd - 2d = a + 2pd - d
Cancelling out the common terms:
2qd - 2d = -d
Rearranging the terms:
2qd = d
Dividing both sides of the equation by 'd':
2q = 1
Simplifying further:
q = 1/2
Now, let's find the (p/2)th term of the AP using the formula:
(p/2)th term = a + ((p/2)-1)d
Substituting the value of 'q' into the equation:
(p/2)th term = a + ((1/2)-1)d
= a - (1/2)d
Comparing this with the (p)th term of the AP:
(p)th term = a + (p-1)d
We can see that the (p/2)th term is half of the (p)th term. Therefore, the correct answer is option 'C': Twice the pth term.
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The sum of (p+q)thand (p-q)th terms of an AP is equal toa)(2p)th termb)(2q)th termc)Twice the pthtermd)Twice the qthtermCorrect answer is option 'C'. Can you explain this answer?
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