In a certain sequence of all positive terms, {a1, a2, a3, …} each term equals the previous term times a constant factor. if (a1)(a5) = 900, what is the value of a3?

Using a geometric sequence, it is found that the value is

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Geometric sequence:

In a geometric sequence, the quotient between consecutive terms is always the same, that is, each term equals the previous term times a constant factor, and this factor is called common ratio.

The nth term of a geometric sequence is given by:

In which is the first term and q is the common ratio.

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Considering :

The 5th term is:

Thus:

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The third term is:

Since

The value is

A similar question is given at link

Let be the constant factor between terms, so that

and so on, with

(notice how the exponent and the subscript add to )

Then

so if

then

Now,

so the third term in the sequence is 30.

the answer is; biodiversity resulting from few ancestors

adaptive radiation occurs when a population is subjected to a large natural environment with spatial ecological variations. the parts of the population in the different ecologies begin to adapt to their local environments. due to non-random mating across the large populations, this causes the different populations from the large population to speciate within their local environments. they, therefore, diverge to different species sharing a common ancestor.