Question

The sum of first n natural numbers was calculated as 1850.But one number was added twice by mistake. What is the difference between n and that number?

Answer 1

Answer:

n is 60 and the number is 20

Hence the difference is 40

Answer 2

Answer:

The difference between n and that number is 40.

Step-by-step explanation:

Given : The sum of first n natural numbers was calculated as 1850.But one number was added twice by mistake.

To find : What is the difference between n and that number?

Solution :

The sum of first n natural number is $S=\frac{n(n+1)}{2}$

First we determine the number of terms,

$1850=\frac{n(n+1)}{2}$

$3700=n^2+n$

$n^2+n-3700=0$

By solving the value of n is 60.33.

So, We take approximately 60 natural numbers whose sum is 1850.

One number was added twice by mistake.

The sum is given by, $S=\frac{n(n+1)}{2}+x$

where, x is the required number.

Now, To find the value of n we multiply the sum by 2 and take square root,

The value of n is

$n=\sqrt{1850\times 2}$

$n=\sqrt{3700}$

$n\approx 60$

Substitute in the formula,

$1850=\frac{60(60+1)}{2}+x$

$1850=\frac{60\times 61}{2}+x$

$1850-\frac{60\times 61}{2}=x$

$x=20$

The value of n is 60.

The required number is 20.

The difference between them is 60-20=40.

Therefore, The difference between n and that number is 40.