Download 1) Use the arithmetic sequence of numbers 2, 4, 6, 8, 10… to find the

1)
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?
Answer: d=2
Show work in this space.
The common difference, d, can be found by subtracting the first term from the
second term
add 2 to each term to arrive at the next term,
or...the difference a2 - a1 is 2.
So d=2
b)
Using the formula for the nth term of an arithmetic sequence, what is 101st
term? Answer: 202
Show work in this space.
To find any term of an arithmetic sequence:
where a1 is the first term of the sequence , d is the common difference, n is the
number of the term to find.
n = 101; a1 =2, d = 2
a101=2+(101-1)2
a 101 = 2+(100)2 =202
So the 101st term is 202
c)
Using the formula for the sum of an arithmetic sequence, what is the sum of
the first 20 terms?
Answer: Sn =420
Show work in this space.
The common difference d for the sequence is 2.
The first term a is 2.
The number of terms is 20.
Substituting a = 2, d = 2, n = 20 in the formula for the sum of n terms
Sn 
20
(2 * 2  (20  1)2)  420
2
Sn =420
d)
Using the formula for the sum of an arithmetic sequence, what is the sum of
the first 30 terms?
Answer: Sn =930
Show work in this space.
a=2, d=2 and n=30
S n
30
(2(2)  (30  1)2  930
2
Sn =930
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.)?
Answer:
2+4 = 6
4+6 =10
6+8 =14
8+10=18
Every sum is larger than the previous and the difference between sums is growing
by4
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: r=3
Show work in this space.
The common ratio, r, can be found by dividing the second term by the first term
Here second term=3 and first term =1
so r=
3
3
1
The ratio r=3
b)
Using the formula for the nth term of a geometric sequence, what is the 10th
term?
Answer: 19683
Show work in this space.
The nth term in this geometric series is (assuming the first term is called n=1):
r(n-1), where r is the ratio (3 in our case)
3(n-1)
Use n = 10:
3(10-1)
39=19683
The 10th term is 19683
c)
Using the formula for the sum of a geometric sequence, what is the sum of the
first 10 terms?
Answer: 29524
Show work in this space.
S = 1(1-310)/(1-3) = -59048/-2 = 29524
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?
Answer: r = 1/3
Show work in this space.
Each term is the previous term divided by 3 or times 1/3
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.
Answer: 1.4999745
Show work in this space.
S = 1(1 - (1/3)10)/(1 - (1/3)) = (1 - 1/59049)/(2/3) = 1.4999745
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.
Answer: 1.4999971
Show work in this space.
S = 1(1 - (1/3)12)/(1 - (1/3)) = (1 - 1/531441)/(2/3) = 1.4999971
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?
Answer: It appears that the sum will always be smaller than 1.5
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 grain of wheat on the first square. Then place two grains of wheat on the next square.
Then place four grains on the third square. Continue this until all 64 squares are covered with
grains of wheat.” As he had just harvested his wheat, Mr. Crane did not consider this much of
an award, but he soon realized he made a miscalculation on the amount of wheat involved.
a)
How much wheat would Mr. Crane have to put on the 24th square?
Answer: 8388608
Show work in this space.
He would put 223 grains on the 24th square
223 =8388608
b)
How much total grain would the traveling salesman receive if the checkerboard
only had 24 squares?
Answer: 16777215
Show work in this space.
The sum of 1 + 2 + 4 + 8 + … + 16777216 = 1(1 - (2)24)/(1 - 2) = 16777215
c)
Calculate the amount of wheat necessary to fill the whole checkerboard (64
squares). How much wheat would the farmer need to give the salesman? Please
provide the answer in either scientific notation, or calculate and show all 20 digits.
Answer: The answer is 2^64 - 1 = 1.844674407 X 1019