If m ≠ n and the sequences p , a , b , q and p , m , n , q each are in AP, then b − a n − m is ___. Options: A . &nbsp 2 3 B . &nbsp 3 2 C . &nbsp 1 D . &nbsp 3 4

Answer: Option C : C Given, p , a , b , q are in AP. ∴ 2 a = p + b ⇒ 2 a − b = p . . . . ( i ) Also, 2 b = a + q . ⇒ 2 b − a = q . . . . ( i i ) Also given that p , m , n , q are in AP. ∴ 2 m = p + n ⇒ 2 m − n = p . . . ( i i i ) Also, 2 n = m + q . ⇒ 2 n − m = q . . . ( i v ) From eqns. ( i ) & ( i i i ) , we get 2 a − b = 2 m − n . . . . ( v ) From eqns. ( i i ) & ( i v ) , we get 2 b − a = 2 n − m . . . . ( v i ) Subtracting ( v i ) from ( v ) , we get 2 a − b − ( 2 b − a ) = 2 m − n − ( 2 n − m ) . ⇒ 2 a − b − 2 b + a = 2 m − n − 2 n + m 3 a − 3 b = 3 m − 3 n b − a = n − m ⇒ b − a n − m = 1

If m≠n and the sequences p,a,b,q and p,m,n,q each are in AP, then b

Discussion Forum : Sequences And Series

Question -

If $m \neq n$ and the sequences $p, a, b, q$ and $p, m, n, q$ each are in AP, then $\frac{b - a}{n - m}$ is ___.

Options:

A .

\frac{2}{3}

B .

\frac{3}{2}

C . 1

D .

\frac{3}{4}

Answer: Option C
:
C
Given,

p, a, b, q

are in AP.

∴ 2 a = p + b

\Rightarrow 2 a - b = p . . . . (i)

Also, 2 b = a + q .

\Rightarrow 2 b - a = q . . . . (i i)

Also given that

p, m, n, q

are in AP.

∴ 2 m = p + n

\Rightarrow 2 m - n = p . . . (i i i)

Also, 2 n = m + q .

\Rightarrow 2 n - m = q . . . (i v)

From eqns.

(i)

(i i i),

we get

2 a - b = 2 m - n . . . . (v)

From eqns.

(i i)

(i v),

we get

2 b - a = 2 n - m . . . . (v i)

Subtracting

(v i)

from

(v),

we get

2 a - b - (2 b - a) = 2 m - n - (2 n - m) .

\Rightarrow 2 a - b - 2 b + a = 2 m - n - 2 n + m

3 a - 3 b = 3 m - 3 n

b - a = n - m

\Rightarrow \frac{b - a}{n - m} = 1

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