if the three consecutive terms of a gp is increased by their middle term then prove that the resulting terms will be in Hp
Answers
Answered by
12
Answer:
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...
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 required.
Answered by
0
Answer:
Step-by-step explanation:
Similar questions