Statement 1 The difference between the sum of the first 100 even natural numbers and the sum of the first 100 odd natural numbers is 100.
Statement 2 The difference between the sum of the first n even natural numbers and sum of the first n odd natural numbers is n.

Options

A. Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1
B. Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1
C. Statement 1 is true, Statement 2 is false
D. Statement 1 is false, Statement 2 is true

Answer

A

Explanation

Statement 1 Sum of first 100 even natural numbers
`E_(1)=2+4+"…"+200=(2(100xx101))/(2)=10100`
Sum of 100 odd natural numbers `=1+3+"….."+199`
`O=(100)/(2)(1+199)=10000`
`:.E-O=100`
So, Statement 1 is true.
Statement 2 Sum of first n natural even numbers `E=2+4+"....."+2n=(2n(n+1))/(2)=n^(2)+n`
Sum of first n odd natural numbers
`O=1+3+"....."+(2n-1)`
`=(n)/(2)[1=2n-1]=n^(2)`
So, `E-O=n^(2)=n-n^(2)=n`
Statement 2 is true and correct explanation for Statement 1.