What is the first term and common ratio for this geometric sequence?

Find the first term and common ratio for the geometric sequence when a_"2" = -15 and a_"4"= -375. I got r=-5, but couldn't seem to get the right value for a1. Can anyone help with this?

1 Answer
Dec 12, 2017

a=3 or a=-3

Explanation:

In geometric sequence, nth term
=ar^(n−1).

So, we have
a_2=ar=-15 ...... [1]
a_4=ar^3=-375 ...... [2]

[2]-:[1] :
(ar^3)/(ar)=(-375)/(-15)
r^2=25
r=5 or r=-5

To find a_1, the first term(a), you just need to plug in r into that formula.

When r=5,
ar=-15
a(5)=-15
a=-3

Also, you can find this using a_4.
a_4=ar^(4-1)=ar^3=-375
a(5)^3=-375
125a=-375
a=-3

We can check it: a_1=-3
a_2=-3*5=-15 [correct]
a_3=-3*5*5=-75
a_4=-3*5*5*5=-375 [correct]
You will see that if a is negative and r is positive, all the terms will be negative.

You find the correct r which is −5. Plug in the r again.

When r=-5,
ar=-15
a(-5)=-15
a=3

Also, you can find this using a_4.
a_4=ar^(4-1)=ar^3=-375
a(-5)^3=-375
-125a=-375
a=3

We can check it: a_1=3
a_2=3*(-5)=-15 [correct]
a_3=3*(-5)*(-5)=75
a_4=3*(-5)*(-5)*(-5)=-375 [correct]
You will see that if a is positive and r is negative, the terms will be +ve, -ve, +ve, -ve.... and so on so on

It is better for you to find a_ using both a_2and a_4 as you can double check if you get the correct a.

Hope this can help you :)