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, #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 :)