Question

A student wanted to find the sum of all the even numbers from 1 to 100. He said:The sum of all the even numbers from 1 to 100 is twice the sum of all the odd numbers from 1 to 100. The sum of all the odd numbers from 1 to 100 is 1002.
Explain why each of these statements is incorrect.

Answer 1

Answer:

The statements are incorrect as: The sum of even numbers from 1 to 100(i.e. 2550) is not double\twice of the sum of odd numbers from 1 to 100(i.e. 2500).

Step-by-step explanation:

We know that sum of an Arithmetic Progression(A.P.) is given by:

$S_{n}=\frac{n}{2}\times (a+a_{n})$

where 'n' denotes the "number" of digits whose sum is to be determined, 'a' denotes the first digit of the series and '$a_{n}$' denote last digit of the series.

Now the sum of even numbers i.e. 2+4+6+8+....+100 is given by the use of sum of the arithmetic progression since the series is an A.P. with a common difference of 2.

$=\frac{50}{2}\times (2+100)$

$=25\times 102\\=2550$

Hence, sum of even numbers from 1 to 100 is 2550.

Also the series of odd numbers is an A.P. with a common difference of 2.

sum of odd numbers from 1 to 100 is given by: 1+3+5+....+99

$=\frac{50}{2}\times (1+99)\\ =25\times 100\\=2500$.

Hence, the sum of all the odd numbers from 1 to 100 is 2500.

Clearly the sum of even numbers from 1 to 100(i.e. 2550) is not double of the sum of odd numbers from 1 to 100(i.e. 2500).

Hence the statement is incorrect.

Answer 2

There would only be 50 odd numbers between 1 and 100 so therefor there would be 50 even so that would only be 50 odd numbers