Geometric series? I just need #1E & 2B,C?

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: 2


Show work in this space.


7-5=2,5-3=2,3-1=2, …………








b) Using the formula for the nth term of an arithmetic sequence, what is 101st term? Answer:201


Show work in this space.2n-1 = 2(201)-1











c) Using the formula for the sum of an arithmetic sequence, what is the sum of the first 20 terms?


Answer:400


Show work in this spacea20=1+(20-1)*2=39 20/2+(1+39)














d) Using the formula for the sum of an arithmetic sequence, what is the sum of the first 30 terms?


Answer:900


Show work in this spacen/2*(a1+a30), a30=1+ (30-1)2=59, 30/2*(1+59)














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:














2) Use the geometric sequence of numbers 1, 2, 4, 8…to find the following:


a) What is r, the ratio between 2 consecutive terms?


Answer:r= ½





Show work in this space.8*1/2=4, 4*1/2=2, 2*1/2=1














b) Using the formula for the nth term of a geometric sequence, what is the 24th term?


Answer:


Show work in this space.














c) Using the formula for the sum of a geometric series, what is the sum of the first 10 terms?


Answer:


Show work in this space

Geometric series? I just need #1E %26amp; 2B,C?
2e)





Sum of the first 1 term: 1


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





See the pattern? {1, 4, 9, 16, 25, 36....}





I'm going to assume you do, but if you REALLY need a hint, think of a TV game show called Hollywood _ _ _ _ _ _ _.





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First off, for 2a), you calculated r to be 1/2. This is wrong.





You calculate r by dividing any term by the term before it:


2/1 = 2


4/2 = 2


8/4 = 2





So actually r = 2.





For 2b), you need the formula for the nth term of a geometric sequence. It is:





a(n) = a(0)rⁿˉ¹





Where a(0) is the first term (1 in this case), and r is the ratio between consecutive terms (2 in this case).





So the 24th term would be:





a(24) = 1(2)^(24 - 1) = 2^23 = 8,388,608





For 2c), all you need is the formula for the sum of a geometric series. It is:





S(n) = a(0)(rⁿ - 1)/(r - 1)





where a(0) is the first term in the sequence, r is the ratio between successive terms, and n is the number of terms.





In this case, a is 1, and you calculated r is still 2.





So:





S(10) = 1(2¹° - 1)/(2 - 1) = (1024 - 1)/1 = 1023.