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Question

If AM between pth and qth terms of an AP be equal to the AM between rth and sth term of the AP, then p+q is equal to

A
r+s
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B
rsr+s
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C
r+srs
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D
r+s+1
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Solution

The correct option is A r+s
We know A.P formula for nth terms with 'a' as the first term and 'd' as the common difference as shown below:

tn=a+(n1)d

Also AM is given between two numbers a and b.
A=a+b2

So arithmetic mean of pth and qth terms of AP is as shown below:

=a+(p1)d+a+(q1)d2

Similarly we can have AM of rth term and sth term of AP as shown below:

=a+(r1)d+a+(s1)d2

Applying the given conditions we get,

a+(p1)d+a+(q1)d2=a+(r1)d+a+(s1)d2

a+pdd+a+qdd2=a+rdd+a+sdd2

a+pdd+a+qdd=a+rdd+a+sdd

2a+d(p+q)2d=2a+d(r+s)2d

d(p+q)=d(r+s)d

p+q=r+s

Hence option A is correct.

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