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Please submit your assignment.

1) Use the arithmetic sequence of numbers 2, 4, 6, 8, 10... to find the following:
a) What is d, the difference between any 2 terms?
Answer:
Show work in this space.

A1,A2,......An.....An+1 be the members in Arithmetic progression
Then d is the common difference
ie d = An+1-An
Here d =2

b) Using the formula for the nth term of an arithmetic sequence, what is 101st term?
Answer:
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The n th term of a AP is given by the formula
An = a + (n-1)d
A101 = a + 100 d

=2 + 100*2= 202

c) Using the formula for the sum of an arithmetic series, what is the sum of the first 20 terms?
Answer:
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Here = 10[2*2+19*2] = 420

Here a denotes the first term and d the common difference

d) Using the formula for the sum of an arithmetic series, what is the sum of the first 30 terms?
Answer:
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=
= 15[2*2+29*2]=930

e) What observation can you make about the successive partial sums of this series (HINT: It would be beneficial to find a few more sums like the sum of the first 2, then the first 3, etc.)?
Answer:

Sum of first two =6
Sum of first three =12
Sum of first four = 20
Sum of first five = 30
...
...

Sum of first n

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:
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Common ratio is 3/1 = 9/3 = 27/9 =3

b) Using the formula for the nth term of a geometric sequence, what is the 10th term?
Answer:
Show work in this space.

Let Xn be the nth term of a geometric progression with first term a and common ratio r then Xn =arn-1 = 1*39= 19683

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.

The general formula for the sum of n terms of a GP is , where a -the first term and r the common ratio
Thus =

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:
Show work in this space.

The common ratio is

b) Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 10 terms? Carry all calculations to 6 decimals on all assignments.
Answer:
Show work in this space.

The general formula for the sum of n terms of a GP is , where a -the first term and r the common ratio
Thus . Here a = 1 and r = 1/3 Thus

=1.499999

c) Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 12 terms? Carry all calculations to 6 decimals on all assignments.
Answer:
Show work in this space.

The general formula for the sum of n terms of a GP is , where a -the first term and r the common ratio
Thus . Here a = 1 and r = 1/3 Thus

= 1.499999

d) What observation can make about the successive partial sums of this series? In particular, what number does it appear that the sum will always be smaller than?
Answer:

Sum of 2 terms 1.44444
Sum of 3 terms 1.48148
Sum of 4 terms 1.49383
Sum of 5terms 1.49794
Sum of 6 terms 1.49931

The sum is always less 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 24nd square?
Answer:
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This problem can be treated as a particular case of geometric progression. Here the first term is one ( one grain) and the common ratio is two. The 23 nd term of this sequence is 1x2x2x2x ......23 times = 223 = 8388608

b) How much total grain would the traveling salesman receive if the checkerboard only had 24 squares?
Answer:
Show work in this space.

In order to fill 24 columns we need
1+2+22+23+24+25+ ....+ 223 .
Using the formula = = 16777216-1 = 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:

Amount of wheat needed to fill 64 squares is S64
Using the formula = = 18446744073709551616 -1
= 18446744073709551615