Question 1085377: 1) Find the 11th term of this sequence. -10,20,-40,80,...
2) Find the next two terms of the following sequence: 14,38,74,122,182,254,...
3) Find the next two terms of the following sequence: -21,-9,11,39,...
4) What is the 9th term in the geometric sequence in which a3 is 36 and a6 is 972?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! I'll do the first two problems to get you started. In the future, please post one problem at a time.
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Problem 1)
This is a geometric sequence since the ratio of terms are all equal to -2
(term 2)/(term 1) = (20)/(-10) = -2
(term 3)/(term 2) = (-40)/(20) = -2
(term 4)/(term 3) = (80)/(-40) = -2
We could keep multiplying each term by -2 to generate enough terms to get to term 11, but that is a bit more work than needed.
Instead, let's form the nth term formula to get
Note how  is the first term and  is the common ratio previously found.
Now plug  into the nth term formula
The 11th term is -10240
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Problem 2)
At first glance, this sequence doesn't seem to have a pattern. It's certainly not arithmetic since
(term 2) - (term 1) = 38 - 14 = 24
(term 3) - (term 2) = 74 - 38 = 36
the differences aren't the same meaning we don't have a common difference.
It's also not geometric because
(term 2)/(term 1) = 38/14 = 2.71 (approx)
(term 3)/(term 2) = 74/38 = 1.95 (approx)
meaning that there isn't a common ratio either.
It turns out that this sequence is following a quadratic progression. The first level of differences (differences between adjacent terms) are
24, 36, 48, 60, 72
which are the results of subtracting each previous term off from its next term. Now focus on the sequence of differences. This sequence {24, 36, 48, 60, 72, ...} is arithmetic with common difference d = 12. This second level difference implies that we have a quadratic progression.
I'm skipping a few steps but we can use technology to construct the equation to be  where x is the term number and y is the term itself.
Plug in  to get
Repeat for 
Therefore the next two terms are 434 and 542
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