Arithmetic and Geometric Sequences
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Use the arithmetic sequence of numbers 2, 4, 6, 8, 10... to find the following:
a) What is d, the difference between any two consecutive terms?
b) Using the formula for the nth term of an arithmetic sequence, what is 101st term?
c) Using the formula for the sum of an arithmetic sequence, what is the sum of the first 20 terms?
d) Using the formula for the sum of an arithmetic sequence, what is the sum of the first 30 terms?
e) What observation can you make about the successive partial 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.)?
2) Use the geometric sequence of numbers 1, 3, 9, 27, ... to find the following:
a) What is r, the ratio between 2 consecutive terms?
b) Using the formula for the nth term of a geometric sequence, what is the 10th term?
c) Using the formula for the sum of a geometric sequence, what is the sum of the first 10 terms?
3) Use the geometric sequence of numbers 1, 1/3, 1/9 , 1/27... to find the following:
a) What is r, the ratio between 2 consecutive terms?
b) Using the formula for the sum of the first n terms of a geometric sequence, what is the sum of the first 10 terms? Carry all calculations to 6 decimals on all assignments.
c) Using the formula for the sum of the first n terms of a geometric sequence, what is the sum of the first 12 terms? Carry all calculations to 6 decimals on all assignments.
d) What observation can make about the successive partial sums of this sequence? In particular, what number does it appear that the sum will always be smaller than?
4) CLASSIC PROBLEM - A traveling salesman (selling shoes) stops at a farm in the Midwest. Before he could knock on the door, he noticed an old truck on fire. He rushed over and pulled a young lady out of the flaming truck. Farmer Crane came out and gratefully thanked the traveling salesman for saving his daughter's life. Mr. Crane insisted on giving the man an award for his heroism.
So, the salesman said, "If you insist, I do not want much. Get your checkerboard and place one penny on the first square. Then place two pennies on the next square. Then place four pennies on the third square. Continue this until all 64 squares are covered with pennies." As he'd been saving pennies for over 25 years, Mr. Brown did not consider this much of an award, but soon realized he made a miscalculation on the amount of money involved.
a) How much money would Mr. Crane have to put on the 32nd square?
b) Calculate the amount of money necessary to fill the whole checkerboard (64 squares). How much money would the farmer need to give the salesman?
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Arithmetic and geometric sequences are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.
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1) Use the arithmetic series of numbers 2 , 4 , 6 , 8 ...to find the following:
a) What is d, the difference between any 2 terms?
Answer:
The difference between any two terms = ( n+1 )th term - ( n )th term
Hence , difference , d = 4-2 = 6 - 4 = 8 - 6 = 2
Answer d = 2
b) Using the formula for the nth term of an arithmetic series, what is 101st term? Answer:
nth term=a+(n-1)d..............(1)
Given a=first term=1
n=101
d=2
putting above values in formula 1
nth term=1+(101-1)*2
=1+200=201
Therefore , nth term = 201
Using the formula for the sum of an arithmetic series, what is the sum of the first 20 terms?
Answer:
formula for the sum of an arithmetic series=n/2(2a +(n-1)d)..............(1)
given that , n = 20
a = first term = 1
Therefore , sum = 20/2(2*1+19*2)
=10*(40)
sum = 400
2) Use the geometric sequence of numbers 1, 3, 9, 27, ... to find the following:
a) What is r, the ratio between 2 consecutive terms?
Answer:
Show work in this space.
As we know that r = common ratio
= ( n+1 ) th term / nth term
...
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