Question

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?

Answer 1

`a=3 or a=-3`

Explanation:

In geometric sequence, `n`th 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_2`and `a_4` as you can double check if you get the correct `a`.

Hope this can help you :)