Question 74369: 1) Use the arithmetic sequence of numbers 1, 3, 5, 7, 9,…to find the following:
a) What is d, the difference between any 2 terms?
Answer:
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b) Using the formula for the nth term of an arithmetic sequence, what is 101st term? Answer:
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c) Using the formula for the sum of an arithmetic sequence, what is the sum of the first 20 terms?
Answer:
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d) Using the formula for the sum of an arithmetic sequence, what is the sum of the first 30 terms?
Answer:
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e) What observation can you make about these sums of this sequence (HINT: It would be beneficial to find a few more sums like the sum of the first 2, then the first 3, etc.)? Express your observations as a general formula in "n."
Answer:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! a)
The difference is the factor between each term. So going from 1 to 3, 3 to 5, 5 to 7, you see that its adding 2 each time. To verify, pick one term and subtract the previous term from it. So lets say I choose 7: I'm going to subtract 5 from it to get a difference of 2. If I pick 5, and subtract 3, I get a difference of 2.So the difference is: d=2
b)
Using what we found earlier, I know that the sequence counts up by 2 each term. So if I'm at 1 (the 1st term) and I go to 3, this means I increase by 2 each term. If I let n=0 then the term is 1, and if I let n=1 then the term is 3. This basically tells me that the arithmetic sequence is 2n+1. To verify, simply plug in the 1st term (n=0) and you'll get 1. Plug in the 2nd term (n=1) you'll get 3, if I let n=2 I get 5, etc. If I wanted to know the 101st term, let n=100 (zero is the first term) and it comes to
 So the 101st term is 201
c)
Using the sum of arithmetic series formula:
 a[1]=first term, a[n]=nth term (ending term which is the 20th term), and n is the number of terms
 Plug in values
 Simplify
 So the sum of the first 20 terms is 400.
d)
Again using the same formula
a[1]=first term, a[n]=nth term (ending term which is the 30th term), and n is the number of terms
 Plug in values
 Simplify
e)
This one is a little tricky to explain, but the series (partial sums) follow the sequence  (notice how each sum is a perfect square). To make a long story short, this sequence is the sum of odd integers. If you want the derivation check out
http://planetmath.org/encyclopedia/SumOfOddNumbers.html
Hope this helps.
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