First slide

Means - A.M/G.M/H.M

Question

If a, b, c are in A.P. and p is the A.M. between a and b and q is the A.M. between b and c, then

Moderate

A

a is the A.M. between p and q
B

b is the A.M. between p and q
C

c is the A.M. between p and q
D

None of these

Solution

$∵ a, b, c are AP ., ∴ 2 b = a + c (1)$
$∵$ p is the A.M. between a and b
$∵$ $p = \frac{a + b}{2}$ (2)
$∵$ q is the A.M. between b and c
$∵$ $q = \frac{b + c}{2}$ (3)
Adding Eq. (2) and (3)
$\begin{array}{l} p + q = \frac{a + b}{2} + \frac{b + c}{2} = \frac{a + c + 2 b}{2} \\ = \frac{2 b + 2 b}{2} = 2 b \end{array}$ [Using (1)]
$∴ 2 b = p + q or b = \frac{p + q}{2}$
Hence, b is the A.M. between p and q.

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