Question

if the three consecutive terms of a G.P. be increased by their middle term, then prove that the resulting terms will be in H.P.​

Answer 1

Answer:

Step-by-step explanation:

Let a be the first term of the GP and r the common ratio.  Then the three terms of the GP are a, ar, ar².

Adding ar (the middle term) to each of these gives:

a+ar, ar+ar, ar²+ar                      ... (1)

or equivalently

a ( 1 + r ),  2ar,  ar ( 1 + r ).             ... (1*)

To show that these are in HP, we need to show that reciprocals

1 / a(1 + r),  1 / 2ar,  1 / ar(1 + r)        ... (2)

are in AP.  This follows once we see that the middle term here is the (arithmetic) mean of the other two terms.

Now... 1/2(1/a(1+r) +1/ar(1+r)

1/2*1+r/ar(1+r)

1/2ar

Therefore the terms in sequence (2) are in AP, and so the terms in sequence (1*)  [ or equivalently, in sequence (1) ] are in HP, as require