SOLUTION: Find the missing term or terms in each geometric sequence.
...,4,_,_,108, ...
Algebra.Com
Question 1083522: Find the missing term or terms in each geometric sequence.
...,4,_,_,108, ...
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Answer for the first blank: 12
Answer for the second blank: 36
-----------------------------------------------------------------------------
Explanation:
The first term of this sequence is 4.
The second term is unknown.
While we don't know it, we do know that it's related to the first term.
Specifically it is equal to 4*r where r is the common ratio of this geometric sequence.
In other words,
second term = (common ratio)*(first term)
second term = (r)*(4)
second term = 4*r
Notice how the second term is built on the first term. In general, any given term is dependent on the previous term (as you'll see below).
Similarly,
third term = (common ratio)*(second term)
third term = (r)*(4*r)
third term = 4r^2
and,
fourth term = (common ratio)*(third term)
fourth term = (r)*(4*r^2)
fourth term = 4r^3
--------------------------------
the expression for the fourth term is 4r^3. We are given that the fourth term is also 108. Equate these two expressions and solve for r.
4r^3 = 108
(4r^3)/4 = 108/4
r^3 = 27
CubeRoot(r^3) = CubeRoot(27) ... ** see note below **
r = 3
Note: applying the cube root to both sides is the same as raising both sides to the 1/3 power.
--------------------------------
Now that we know the common ratio is r = 3, we use it to find the missing terms we need.
Plug r = 3 into the second term equation (defined above):
second term = 4*r
second term = 4*3
second term = 12
Do the same for the third term equation as well:
third term = 4r^2
third term = 4*(3)^2
third term = 4*9
third term = 36
An alternative way to compue the third term is:
third term = (common ratio)*(second term)
third term = (r)*(12)
third term = (3)*(12)
third term = 36
and we get the same answer
--------------------------------
Side Notes:
Plugging r = 3 into the equation for the fourth term leads to this:
fourth term = 4r^3
fourth term = 4*(3)^3
fourth term = 4*27
fourth term = 108
which helps verify that we have the right common ratio (r).
Another way to see if we have the right common ratio is to divide each term (but the first) over its previous term. Doing so yields
(second term)/(first term) = 12/4 = 3
(third term)/(second term) = 36/12 = 3
(fourth term)/(third term) = 108/36 = 3
We get 3 each time. This is another way to confirm that we have the right common ratio (r).
RELATED QUESTIONS
find the missing terms for the geometric sequence
..., 4, ___, ___, 108,... (answered by ikleyn)
X=1,2,3,4,5
Y=2,_,_,_,162
For each geometric, find the missing term in the... (answered by josgarithmetic)
Find the missing term in geometric sequence.... (answered by ikleyn)
The first term of the sequence is -2 and the fifth term is -1/8. List 4 things you know.... (answered by richwmiller)
Find the missing term of each geometric sequence?
29. 1, [?] , 49
30. 3/4, [?] ,... (answered by user_dude2008)
Find the missing term of each geometric sequence?
26. 16, [?] , 4
27. 25, [?] , 225
(answered by Mathtut)
What is the missing term of the geometric sequence... (answered by math_helper,Alan3354,ikleyn,MathTherapy)
The geometric mean between the first two terms in a geometric sequence is 32. If the... (answered by rothauserc)
What are the four missing terms between 4 and 1/16 in the geometric... (answered by stanbon)