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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+sWe know A.P formula for nth terms with 'a' as the first term and 'd' as the common difference as shown below:tn=a+(n−1)dAlso AM is given between two numbers a and b. A=a+b2So arithmetic mean of pth and qth terms of AP is as shown below:=a+(p−1)d+a+(q−1)d2Similarly we can have AM of rth term and sth term of AP as shown below:=a+(r−1)d+a+(s−1)d2Applying the given conditions we get,a+(p−1)d+a+(q−1)d2=a+(r−1)d+a+(s−1)d2 a+pd−d+a+qd−d2=a+rd−d+a+sd−d2a+pd−d+a+qd−d=a+rd−d+a+sd−d 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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