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arithmetic and geometric progression , classical problem

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I need assistance in discerning sequences and series including the classical grains and checkboard problem

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https://brainmass.com/math/algebraic-geometry/arithmetic-geometric-progression-classical-problem-65625

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1) Use the arithmetic series of numbers 1, 3, 5, 7, 9,...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 = 3-1 = 5-3 = 7-5 = 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

Using the formula for the sum of an arithmetic series, what is the sum of the first 30 terms?
Answer:
formula for the sum of an arithmetic series=n/2(2a +(n-1)d)..............(1)
given that n = 30
a = first term =1
= 30/2(2*1+29*2)
...

Solution Summary

The solution contains answers to various questions involving A.P and G.P concepts including the famous classical problem involving checkboard and grains.

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Sequences and Series

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?
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 series, what is the sum of the first 20 terms?
d) Using the formula for the sum of an arithmetic series, what is the sum of the first 30 terms?
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.)?

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 series, 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 series, 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 series, 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 series? 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 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?
b) How much total grain would the traveling salesman receive if the checkerboard only had 24 squares?
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.

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