Subjective Type

If $$a , b ,c$$ are in G . P and $$a -b , a - c$$ and $$b - c$$ are in H . P . then prove that $$a + 4b + c$$ is equal $$0$$

Solution

a , b , c , in G . P$$ \Rightarrow$$ b = ar , c = $$ar^2$$
$$\therefore \, \, -2r \, = \, (r \, + \, 1 ) ^2 \, or \, r^2 \, + \, 4r \, + \, 1 \, = 0 $$
now $$ c \, + \, 4b \, + \, a = \, a (\, r^2 \, + \, 4r \, + \, 1 \, = 0 ) $$

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