Question

The sum of all integers from -23 to N is N. What is the value of N?

Answer 1

Answer:24

Step-by-step explanation:

Notice that it is in AP

where first term = - 23

Last term = N

No. Of terms(n)= N-(-23) +1

Sum of AP = n/2 × ( 1st term + last term)

Sum = (N+ 24)(N-23)/2

BTP,

(N+ 24)(N-23)/2 = N

Solving we get,

N= 24

Answer 2

The value of N is 24.

Given:

The sum of all integers from -23 to N is N.

To Find:

The value of N.

Solution:

To find the value of N we will follow the following steps:

According to the question:

Integers are from -23 to N.

So,

-23, -22, -21,.........,(N-1), N

It is forming an AP.

So,

The first term = -23

Common difference (d) = difference of two consecutive terms = -22 - (-23) = -22+23 = 1

Now,

The sum of numbers is calculated by the formula =

$\frac{n}{2} (first \: term + last \: term)$

Here, n is the total number of terms.

To find the total numbers we will use a formula to find the nth term.

Nth term is calculated by the formula

$N = - 23 + (n - 1)1$

$N + 23 = n - 1$

$N + 24 \: = n$

N+24 is the total number of terms.

Now,

Putting values in the above formula of finding the sum of all the numbers from -23 to N.

We get,

$sum = \frac{N + 24}{2} ( - 23 + N ) = \frac{(N + 24)(N -23)}{2}$

It is given that sum is N.

So,

$sum = \frac{N + 24}{2} ( - 23 + N ) = N$ $(N + 24)(N -23) = 2N$

N² - 22N + 24N - 552 = 2N

N² - 552 = 0

N = √552 = 23.49

N is an integer so, cannot be in fractions.

N = 24

Henceforth, The value of N is 24.

#SPJ2