Question

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?

Answer 1

Let $k$ be the constant factor between terms, so that

$a_2=ka_1$

$a_3=ka_2=k^2a_1$

and so on, with

$a_n=ka_{n-1}=\cdots=k^{n-1}a_1$

(notice how the exponent and the subscript add to $n$)

Then

$a_5=k^4a_1$

so if

$a_1a_5=900$

then

$k^4{a_1}^2=900\implies k^4=\dfrac{900}{{a_1}^2}=\left(\dfrac{30}{a_1}\right)^2\implies k^2=\dfrac{30}{a_1}$

Now,

$a_3=k^2a_1=\dfrac{30a_1}{a_1}=30$

so the third term in the sequence is 30.