Question 1085377
<font color="black" face="times" size="3">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 


{{{a[n] = a*(r)^(n-1)}}}
{{{a[n] = -10*(-2)^(n-1)}}}


Note how {{{a = -10}}} is the first term and {{{r = -2}}} is the common ratio previously found.


Now plug {{{n = 11}}} into the nth term formula


{{{a[n] = -10*(-2)^(n-1)}}}
{{{a[11] = -10*(-2)^(11-1)}}}
{{{a[11] = -10*(-2)^(10)}}}
{{{a[11] = -10*(1024)}}}
{{{a[11] = -10240}}}


The 11th term is <font color=red>-10240</font>


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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 <a href="http://wtmaths.com/quadratic_progressions.html">quadratic progression</a>. 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 {{{y = 6x^2 + 18x + 14}}} where x is the term number and y is the term itself. 


Plug in {{{x = 7}}} to get


{{{y = 6x^2 + 18x + 14}}}
{{{y = 6(7)^2 + 18(7) + 14}}}
{{{y = 434}}}


Repeat for {{{x = 8}}}


{{{y = 6x^2 + 18x + 14}}}
{{{y = 6(8)^2 + 18(8) + 14}}}
{{{y = 542}}}


Therefore the next two terms are <font color=red>434 and 542</font></font>