# How do you write a rule for the nth term of the geometric term given the two terms #a_3=24, a_5=96#?

##### 1 Answer

#### Answer:

#### Explanation:

The general formula for a geometric sequence is

I'm going to explain how to do this problem two ways.

**The Long Way**

Since we are given

#a_3 = 24# and#a_5 = 96# , we can substitute them into the formula.

#a_3 = a_1 * r^(3-1) #

# a_3 = a_1 * r^2#

#color(blue)(24 = a_1 * r^2)#

#a_5 = a_1 * r^(5-1) #

# a_5 = a_1 * r^4#

#color(blue)(96= a_1 * r^4)# Now we can solve the system of equations:

#color(blue)(24 = a_1 * r^2)# #-># solve for#a_1#

#a_1=24/r^2#

#color(blue)(96= a_1 * r^4)#

#96=24/r^2 * r^4# #-># substitute the value of#a_1# into the second equation

#96=24 * r^2#

#4=r^2#

#r=+-2# Now that we have the value of

#r# , we can find the value of#a_1# . Using the first equation,#color(blue)(24 = a_1 * r^2)# , we get

#24 = a_1 * r^2#

#24 = a_1 * (+-2)^2#

#24 = a_1 * 4#

#a_1=6# So our formula for the sequence can be either

#color(red)(a_n = 6 * 2^(n-1))# or#color(red)(a_n = 6 * (-2)^(n-1))# .To verify if these are correct, you can write out the first few terms and see if they match the information given in the problem.

#color(red)(a_n = 6 * 2^(n-1)) # The common ratio is

#2# , so start with#6# and multiply each term by#2 => 6, 12, 24, 48, 96#

#color(red)(a_n = 6 * (-2)^(n-1))# The common ratio is

#-2# , so start with#6# and multiply each term by#-2 => 6, -12, 24, -48, 96# In both of these formulas,

#a_3=24# and#a_5=96# .

**The Short Way**

We are given

#a_3# and#a_5# , so we can easily find out#a_4# in order to get the value of#r# .

#a_3, a_4, a_5#

#24, a_4, 96# To find

#a_4# , we can simply calculate the geometric mean.

#(a_4)^2 = 24 * 96 => a_4 = +-sqrt(24 * 96) = +-sqrt2304 = +-48# So the three terms are either

#24, 48, 96# , meaning that#r = 48/24 = 2# , or the terms are#24, -48, 96# , meaning that#r=-48/24 = -2# .After you find

#r# , you can find#a_1# the same way we did above. In the end, you get#color(red)(a_n = 6 * 2^(n-1))# or#color(red)(a_n = 6 * (-2)^(n-1))# .

(This method is easier in the context of this problem, but if you were given terms such as