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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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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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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)
...

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