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Harmonic Progression

If the A.M. b...

Question

If the $A . M .$ between $p^{t h}$ and $q^{t h}$ terms of an $A . P .$ is equal to the $A . M .$ between $r^{t h}$ and $s^{t h}$ terms of the $A . P .$ , then which of the following is true

p r = q s

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p q = r s

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p + r = q + s

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p + q = r + s

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Solution

The correct option is D $p + q = r + s$
Given $\frac{1}{2} (t_{p} + t_{q}) = \frac{1}{2} (t_{r} + t_{s}) \Rightarrow a + (p - 1) d + a + (q - 1) d = a + (r - 1) d + a + (s - 1) d \Rightarrow (p - 1 + q - 1) d = (r - 1 + s - 1) d \Rightarrow p + q = r + s$

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Similar questions

If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p - q, q - r, r - s are in G.P.

Q. If A.M. between

p^{t h}

and

q^{t h}

terms of an A.P. be equal to the A.M. between

r^{t h}

and

s^{t h}

terms of the A.P., then prove that

p + q = r + s

Q. If the

A . M .

between

p^{t h}

and

q^{t h}

terms of an

A . P .

is equal to the

A . M .

between

r^{t h}

and

s^{t h}

terms of the

A . P .

, then which of the following is true

Q. If AM between

p^{t h}

and

q^{t h}

terms of an AP be equal to the AM between

r^{t h}

and

s^{t h}

term of the AP, then

p + q

is equal to

Q. If the

p t h

q t h

r t h

and

s t h

terms of an

A . P .

are in

G . P .

, then

p - q, q - r, r - s

are in

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Harmonic Progression

Standard XII Mathematics

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If the A.M. between pth and qth terms of an A.P. is equal to the A.M. between rth and sth terms of the A.P., then which of the following is true

The correct option is D p+q=r+sGiven 12(tp+tq)=12(tr+ts)⇒a+(p−1)d+a+(q−1)d=a+(r−1)d+a+(s−1)d⇒(p−1+q−1)d=(r−1+s−1)d⇒p+q=r+s

If the $A . M .$ between $p^{t h}$ and $q^{t h}$ terms of an $A . P .$ is equal to the $A . M .$ between $r^{t h}$ and $s^{t h}$ terms of the $A . P .$ , then which of the following is true

The correct option is D $p + q = r + s$
Given $\frac{1}{2} (t_{p} + t_{q}) = \frac{1}{2} (t_{r} + t_{s}) \Rightarrow a + (p - 1) d + a + (q - 1) d = a + (r - 1) d + a + (s - 1) d \Rightarrow (p - 1 + q - 1) d = (r - 1 + s - 1) d \Rightarrow p + q = r + s$