SOLUTION: Find the 100th and nth term for each of the following sequences. a. 1,5,9,13,... b.70,120,170... c.1,3,9.... d.8,8^4,8^7,8^10,... e.109+7x3^32,109+8x3^32,109+9x3^32,...

Algebra ->  Sequences-and-series -> SOLUTION: Find the 100th and nth term for each of the following sequences. a. 1,5,9,13,... b.70,120,170... c.1,3,9.... d.8,8^4,8^7,8^10,... e.109+7x3^32,109+8x3^32,109+9x3^32,...      Log On


   

Question 1151584: Find the 100th and nth term for each of the following sequences.
a. 1,5,9,13,...
b.70,120,170...
c.1,3,9....
d.8,8^4,8^7,8^10,...
e.109+7x3^32,109+8x3^32,109+9x3^32,...
Found 2 solutions by ikleyn, MathLover1:
Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
.

            I will solve and answer parts  (a),  (b)  and  (c)  only. 


a)  is the arithmetic progression with the first term 1 and the common difference of 4.

    a%5Bn%5D = 1 + 4*(n-1) = 4n - 3.



b)  is the arithmetic progression with the first term 70 and the common difference of 50.

    a%5Bn%5D = 70 + 50*(n-1) = 50n + 20.



c)  is the geometric progression with the first term 1 and the common ratio of 3.

    a%5Bn%5D = 1%2A3%5E%28n-1%29 = 3%5E%28n-1%29.


I just opened your eyes --- you do the rest.

----------------

This problem is about the  SIMPLEST  properties of arithmetic and geometric progressions.

Learn the subjects from these two groups of  introductory  lessons in this site

    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions

and

    - Geometric progressions
    - The proofs of the formulas for geometric progressions
    - Problems on geometric progressions
    - Word problems on geometric progressions


Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topics  "Arithmetic progressions"  and  "Geometric progressions".


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Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Find the 100th and nth term for each of the following sequences.
a. 1,5,9,13,...
a%5B1%5D=1
a%5B2%5D=5=1%2B4
a%5B3%5D=9=5%2B4
a%5B4%5D=13=9%2B4
=> common difference d=4, arithmetic sequence

the nth formula:
a%5Bn%5D=+a%5B1%5D%2B%28n-1%29d
so, the 100th term will be:
a%5B100%5D=+1%2B%28100-1%294
a%5B100%5D=+1%2B99%2A4
a%5B100%5D=+1%2B396
a%5B100%5D=+397


b.70,120,170...
a%5B1%5D=70
a%5B2%5D=120=70%2B50
a%5B3%5D=170=120%2B50
=> common difference d=50, arithmetic sequence

the nth formula:
a%5Bn%5D=+a%5B1%5D%2B%28n-1%29d
so, the 100th term will be:
a%5B100%5D=+70%2B%28100-1%2950
a%5B100%5D=+70%2B99%2A50
a%5B100%5D=+70%2B4950
a%5B100%5D=+5020

c.1,3,9....
a%5B1%5D=1
a%5B2%5D=3=1%2A3
a%5B3%5D=9=1%2A3%5E2
=> common ratio r=3, geometric sequence
the nth formula:
a%5Bn%5D+=+a%5B1%5D%2A3%5E%28n+-+1%29
the 100th term will be:
a%5B100%5D+=+1%2A3%5E%28100+-+1%29
a%5B100%5D+=+1%2A3%5E99
a%5B100%5D+=+171792506910670443678820376588540424234035840667

d.
8,8^4,8^7,8^10,...
note that exponents of first and second term differ by 3, same for second and third term, and same for third and fourth term
a%5B1%5D=8%5E1
a%5B2%5D=8%5E4=8%5E%281%2B3%29
a%5B2%5D=8%5E7=8%5E%284%2B3%29

the nth formula:
a%5Bn%5D+=8%5E%283n+-+2%29 where n=0,1,....
the 100th term will be:
a%5B100%5D+=8%5E%283%2A100+-+2%29
a%5B100%5D+=8%5E%283%2A98%29
a%5B100%5D+=8%5E294 => leave like this because this number is really big

e.
109+7x3^32,109+8x3^32,109+9x3^32,...
I am not quite sure if x represents, I assume multiplication

109%2B7%2A3%5E32,109%2B8%2A3%5E32,109%2B9%2A3%5E32

only difference is in numbers 7,8,9

generally, %28n%2B7%29 where n= 0,1,....

the nth formula:
a%5Bn%5D+=109%2B%28n%2B7%29%2A3%5E32
a%5B100%5D+=109%2B%28100%2B7%29%2A3%5E32
a%5B100%5D+=109%2B107%2A3%5E32+
a%5B100%5D+=198273160207147096