Question 84127: 1. Use the arithmetic sequence of numbers 1,3,5,7,9,....to find the following:
a. What is d, the difference between two consecutive terms?
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**I am not sure what the teacher is asking here, but I think the answer is 1. 1+2=3+1=4+1=5 and so forth.
b. Using the formula for the n(th)term of an arithmetic sequence, what is the 101(st) term?
show your work
**Sn=n/2(a1+an) **I got 10,150.50
c.Using the formula for the sum of an arithmetic sequence, what is the sum of the first 20 terms?
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**I got 400***
d.Using the formula for the sum of an arithmetic sequence, what is the sum of the first 30 terms?
show work
***I got 939***
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"
**I think the answer here is that the sequence of numbers goes up by one.
***Can you please help me, I want to make sure these answers are correct before I submit them.
Thank you so much for your time, I really appreciate it,
Jennifer
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
 So the sum of the first 30 terms is 900
e)
Sum of the first 2 terms
1+3=4
Sum of the first 3 terms
1+3+5=9
Sum of the first 4 terms
1+3+5+7=16
Sum of the first 5 terms
1+3+5+7+9=25
Sum of the first 6 terms
1+3+5+7+9+11=36
Sum of the first 7 terms
1+3+5+7+9+11+13=49
Sum of the first 8 terms
1+3+5+7+9+11+13+15=64
Sum of the first 9 terms
1+3+5+7+9+11+13+15+17=81
Sum of the first 10 terms
1+3+5+7+9+11+13+15+17+19=100
Notice how the partial sums are all perfect squares. So the sums follow the sequence 
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