Sequences

Arithmetic and Geometric progressions.

Sequences

Postby Guest » Wed Jun 15, 2016 11:38 am

A lecture theatre has a trapezium shaped floor plan so that the number of chairs in successive rows are in arithmetic sequence.the back rows of chairs contain 8 chairs and the front row contains 30. there are 12 rows altogether. 1.find the number of seats in the theatre 2. find the percentage of seats that are in the rear half of the theatre i cant work this out?
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Re: Sequences

Postby Guest » Wed Jun 15, 2016 8:18 pm

The numbers are fairly simple
nth term = a + d(n-1)
that gives 8 + 11d = 30 ......so 11d = 22 ......so d = 2

So each row has 8, 10 ,12,14,16,18,20,22,24,26,28,30 seats ..... at total of 228
you can also use the formula for the sum of n terms

The back 6 rows have 78 seats total, so percent = 78 / 228 = 34.2 percent.
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Re: Sequences

Postby leesajohnson » Sat Jul 02, 2016 5:28 am

34.5 % of seats that are in the rear half of the theatre.

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Re: Sequences

Postby Mathlibra » Sat Aug 17, 2019 10:30 pm

Sequences
A sequence, or progression, is a set of numbers obeying a simple rule. In a sequence, the successive terms (or numbers) are in definite order and formed according to some rule.
2, 4, 6, 8, 10, ... (1.0)
13, 18, 23, 28, 33, ... etc. (1.1)
Example (1.0) is the sequence of positive even integers. We denote nth term of a general sequence by [tex]u_{n }[/tex], and the sequence by [tex]u_{1 }[/tex], [tex]u_{2 }[/tex], [tex]u_{3 }[/tex], [tex]u_{4 }[/tex], ..., [tex]u_{n }[/tex], [tex]u_{n+1 }[/tex], [tex]u_{n+2 }[/tex], ... . The simple rule defining a general sequence, [tex]u_{n }[/tex], is often given in the form of a function of n, although this is not necessarily so. For examples (1.0) and (1.1), we have [tex]u_{n }[/tex]=2n and [tex]u_{n }[/tex]=5n+8.

Example 1: Find [tex]u_{n }[/tex] in terms of n for the following sequences.
(i) 1, 3, 5, 7, 9, ...
(ii) 1, ⅛, 1/27, 1/64, ...
(iii) -1, 2, -3, 4, -5, ...
(iv) 1, 4, 9, 16, ...
(v) 1, -1, 1, -1, 1, ...
:idea: Solution:
(i) The terms in the sequence can be written as
1+0, 2+1, 3+2, 4+3, 5+4, ...
[tex]u_{n }[/tex]=n+(n-1)=2n-1
(ii) The terms in the sequence can be written as
1/[tex]1^{3}[/tex], 1/[tex]2^{3}[/tex] ,1/[tex]3^{3}[/tex], 1/[tex]4^{3}[/tex], …
[tex]u_{n }[/tex]=1/[tex]n^{3}[/tex]
(iii) Inspecting this sequence, the terms can be written as
1⋅[tex](-1)^{1}[/tex], 2⋅[tex](-1)^{2}[/tex], 3⋅[tex](-1)^{3}[/tex], 4⋅[tex](-1)^{4}[/tex], …
[tex]u_{n }[/tex]=n[tex](-1)^{n}[/tex]
Try (iv) and (v).

A good example of a sequence where [tex]u_{n }[/tex] cannot be written in form of a function of n is the Fibonacci sequence;
[tex]u_{1 }[/tex]=0;
[tex]u_{2 }[/tex]=1;
[tex]u_{n }[/tex]=[tex]u_{n-1 }[/tex]+[tex]u_{n-2 }[/tex]; n≥3
In this type of sequence, there is mathematical relation between consecutive terms. This is called recurrence.

Example 2. Find the 4th term in the sequence with general term 3⋅[tex]5^{n-1}[/tex].
:idea: Solution:
The 4th term in the sequence with general term
3⋅[tex]5^{n-1}[/tex] is 3⋅[tex]5^{4-1}[/tex]=3⋅[tex]5^{3}[/tex]=3⋅125=375.

Read more: Get Terms from Sequences :idea:

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