9. If the A.M. between pth and qth terms of an A.P. be equal to the AM between
mth and nth terms of the A.P., then show that p+q=m+n.
Answers
Answer:
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Step-by-step explanation:
We know A.P formula for nth terms with 'a' as the first term and 'd' as the common difference as shown below:
t
n
=a+(n−1)d
Also AM is given between two numbers a and b.
A=
2
a+b
So arithmetic mean of pth and qth terms of AP is as shown below:
=
2
a+(p−1)d+a+(q−1)d
Similarly we can have AM of rth term and sth term of AP as shown below:
=
2
a+(r−1)d+a+(s−1)d
Applying the given conditions we get,
2
a+(p−1)d+a+(q−1)d
=
2
a+(r−1)d+a+(s−1)d
2
a+pd−d+a+qd−d
=
2
a+rd−d+a+sd−d
a+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.