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Question 658798: 1. A theatre has 49 seats, 53 in the second row, 57 seats in the third row and so on.
a. can the number of seats in each row be numbered in geometric or arithmetic sequence?
b. write the general term for a sequence a[n] that gives the number of seats in row n. a[n]=
c. how many seats are in row 18?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! row 1 = 49
row 2 = 53
row 3 = 57
this is an arithmetic sequence
formula for the last term in an arithmetic sequence is l = a + (n-1)d
a is the first number in the sequence
l is the last number in the sequence
n is the number of terms in the sequence.
for the part of the sequence that is given to you, we get:
a = 49
n = 3
d = 4
use this formula to find the number of the last term in the sequence.
you get l = 49 + 2*4 = 49 + 8 = 57
as you can see, the last term in the sequence is 57 as given so the formula is good.
we know it's an arithmetic sequence because it has a common difference which is 4.
the number of seats in row 18 would be given by the same formula.
l = a + (n-1) * d which becomes l = 49 + 17*4 which becomes l = 49 + 68 which becomes l = 117
if you wanted to take the trouble of actually plotting out the sequence, it would look like this:
i x.i
1 49
2 53
3 57
4 61
5 65
6 69
7 73
8 77
9 81
10 85
11 89
12 93
13 97
14 101
15 105
16 109
17 113
18 117
i is the row number
x.i is the number of seats in that row.
this is not a geometric sequence because you don't have a common ratio.
the formula for a geometric sequence is l = a*r^(n-1)
an example of a geometric sequence would be:
1
2
4
8
the common ratio is 2 because each number is 2 times the number before it.
in this geometric sequence, the 4th number in the sequence would be found using the formula of l = 1*2^3 which becomes l = 1*8 which becomes l = 8.
since the 4th number in the sequence is 8, the formula works.
in your problem there is no common ratio.
53/49 = 1.0816326...
57/53 = 1.0754716...
there is, however, a common difference of 4.
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