An A.P., a G.P. and a H.P. have the same first and last terms and the same odd number of terms. The middle terms of the three series are in
A. A.P.
B. G.P.
C. H.P.
D. None of these

Hint
In a quadratic factorization employing splitting of the middle term, where \[x\] is the product of two factors and last term is the sum of the two factors, total of each phrase in an AP. The eighth term, then, will be the middle term of an arithmetic progression, is the correct response. The two middle terms are the \[n/2\] and \[n/2{\rm{ }} + {\rm{ }}1\] terms if ' \[n\] ' is even. The eighth and ninth words, then, will serve as the progression's middle terms.
A middle term in logic is a term that appears in both premises but not the conclusion of a categorical syllogism (as a subject or predicate of a categorical proposition).
Formula used:
if A.M between a and b is
A.M = \[\frac{{(a + b)}}{2}\]
if G.M between a and b is
G.M = \[\sqrt {ab} \]
if H.M between a and b is
\[\frac{{2ab}}{{(a + b)}}\]
Complete step-by-step solution
Assume A.P., G.P., and H.P.'s first and final terms.
Let the first and last terms of A.P., G.P., and H.P., all of which contain an odd number of words, be a and b.
Now, calculate the centre term of A.P., G.P., and H.P.
The centre term of A.P is \[\frac{{(a + b)}}{2}\] ---(1)
The centre term of G.P is \[\sqrt {ab} \] ---(2)
The centre term of H.P is \[\frac{{2ab}}{{(a + b)}}\] ---(3)
Here, the middle terms of the three series are multiplied to calculate the answer
Equation (1) and (3) are multiplied
\[\frac{{(a + b)}}{2} \times \frac{{2ab}}{{(a + b)}}\]
This then becomes
\[ = > ab = > {(\sqrt {ab} )^2}\]
This is equal to the square value of the center term of G.P.
So, it is proved that all the terms are in G.P.
Therefore, the correct option is B.

Note
AP, GP, and HP stand for the average or mean of the series. Arithmetic Mean, Geometric Mean, and Harmonic Mean, respectively, are denoted by the letters AM, GM, and HM. The abbreviations AP, GP, and HP stand for Arithmetic Progression, Geometric Progression, and Harmonic Progression, respectively. Each next phrase in a geometric progression is obtained by multiplying the common ratio by the term that came before it.
If the reciprocal of the terms is in AP, a series of numbers is referred to as a harmonic progression.