In a certain sequence of all positive terms, {a1, a2, a3, …} each term equals the previous term times a constant factor. If (a1)

Question

1 month ago

6

(a5) = 900, what is the value of a3?

1 answer:

babunello [11.8K] · Answer 1 · 2026-04-06T19:57:24+03:00

7 0

Let $k$ represent the constant multiplier between successive terms, which allows us to express

$a_2=ka_1$

$a_3=ka_2=k^2a_1$

and so forth, given that

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

(observe that the sum of the exponent and the subscript equals $n$ )

Thus

$a_5=k^4a_1$

implying that if

$a_1a_5=900$

then

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

At this point,

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

therefore the third term in the series is 30.