Question 1208641
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Answer: <font color=red>1, 3, 5, 7</font>


Explanation


AP = arithmetic progression
GP = geometric progression


We have these 3 sequences
<table border = "1" cellpadding = "5"><tr><td>Sequence 1</td><td>1</td><td>1</td><td>3</td><td>9</td></tr><tr><td>Sequence 2</td><td>a</td><td>a+d</td><td>a+2d</td><td>a+3d</td></tr><tr><td>Sequence 3</td><td>1+a</td><td>1+a+d</td><td>3+a+2d</td><td>9+a+3d</td></tr></table>
Sequence 3 is the sum of sequence 1 and sequence 2.
Add straight down.


Because we're told that sequence 3 is geometric, dividing any term over its previous term will get us the common ratio r.


r = (2nd term)/(1st term) = (1+a+d)/(1+a)
r = (3rd term)/(2nd term) = (3+a+2d)/(1+a+d) 


Equate those expressions to form this equation
(1+a+d)/(1+a) = (3+a+2d)/(1+a+d)
Solving for 'a' leads to a = 0.5d^2 - 1
I'll let the student do the scratch work.


Furthermore,
r = (3rd term)/(2nd term) = (3+a+2d)/(1+a+d) 
r = (4th term)/(3rd term) = (9+a+3d)/(3+a+2d)


Equating them gives us (3+a+2d)/(1+a+d) = (9+a+3d)/(3+a+2d)


Plug in a = 0.5d^2 - 1 and we get 
(3+0.5d^2 - 1+2d)/(1+0.5d^2 - 1+d) = (9+0.5d^2 - 1+3d)/(3+0.5d^2 - 1+2d)


Solving that equation yields d = 2


Plug d = 2 into a = 0.5d^2 - 1 to get a = 1.


To summarize:
a = 1
d = 2


This table
<table border = "1" cellpadding = "5"><tr><td>Sequence 1</td><td>1</td><td>1</td><td>3</td><td>9</td></tr><tr><td>Sequence 2</td><td>a</td><td>a+d</td><td>a+2d</td><td>a+3d</td></tr><tr><td>Sequence 3</td><td>1+a</td><td>1+a+d</td><td>3+a+2d</td><td>9+a+3d</td></tr></table>
Updates to
<table border = "1" cellpadding = "5"><tr><td>Sequence 1</td><td>1</td><td>1</td><td>3</td><td>9</td></tr><tr><td>Sequence 2</td><td><font color=red>1</font></td><td><font color=red>3</font></td><td><font color=red>5</font></td><td><font color=red>7</font></td></tr><tr><td>Sequence 3</td><td>2</td><td>4</td><td>8</td><td>16</td></tr></table>
Sequence 2 is arithmetic because we add 2 to each term to get the next term.
The nth term of this sequence is 2n-1 where n is an integer that starts at n = 1.


Sequence 3 is geometric since dividing any given term over its previous term results in the same common ratio
4/2 = 2
8/4 = 2
16/8 = 2
Put another way: we double each term to get the next term.
The nth term of this geometric sequence is 2^n.
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