Positive Integers

Algebra Level 1

The average of the first 100 positive integers is $\text{\_\_\_\_\_\_\_\_\_\_}.$

50 50.5 51 101

13 solutions

Mardokay Mosazghi
Jun 4, 2014

No need for calculations. Average = (first number + last number)/2 = (100 + 1)/2 = 101/2 = 50.5

How did you derived this

Archiet Dev - 7 years ago

To derive the total from 1-100 would be as goes: We will switch between using this situation and the general formula. There are 50 pairs of numbers that make 100 (100+0,99+1,98+2...51+49.) We can write this equation as n/2*n or (n^2)/2. However, we also know that there is also 50 which we haven't accounted for, and seeming as there is no pair that adds up with 50 (only one 50), we will have to add on 50 to this total. Therefore total= (n^2)/n + n/2. This equation can be factorized into:

t=n(n+1)/2. Where t is total. If we sub in 100 as n, we will get 5050. This is the total of all the positive integers from 1-100. Now all we have to do this divide by 100 to find out the average (as there are 100 numbers).

Therefore the general formula is

a= n(n+1)/2n. Where a is average. So a= (100*101)/200 which gives 50.5

Jeevith Gnana - 7 years ago

why is the last number 1

murali ch - 3 years ago

Isn't average the sum of all numbers divided by the number of numbers

khush Ratadia - 2 years, 3 months ago

Aditya Gupta
Jan 22, 2017

sum of 1st hundred naturals can be finded by Arthmatic progreesion sum of terms whisch are in A.P=n/2(1st term +last term) Average of hundred natural numer s can be given by sum /100

Punithan Mech
Jun 6, 2014

1 to 49 plus 51 to 99 which gives 4900 and last 100 and remaing number is 50 which sums up 5050 and it is divided by 100 gives the answer 50.5

[n(n+1)]÷2 therefore [100(100+1)]÷2 is 5050. Average= 5050÷100=50,5

Elisha Ayanda SIBANDA - 7 months ago

I didn't understand

Rohan Bobby - 5 years, 8 months ago

Atif Shahid
Dec 6, 2015

I don't why.. but I did doesn't make sense but it worked.. I used the n!=(n+1)/2 formula and it totally worked... can someone explain plz

Aditya Patil
Feb 7, 2021

find the sum of numbers using the formula of sum of arithmetic progression and then divide the answer by 100 since we need to find the average.

Dinasauress Zulaika
Nov 26, 2019

Zero is an integer thus it is included. Thus the total term for all the positive integer for the first 100 is 101. We are finding average. Thus we take the total term and divide with 2

Harry Biswas
Dec 2, 2017

Everyone knows that, (n+1)/2 here n=100 since (100+1)/2=50.5(ans)

Preeti Parashar
Nov 15, 2017

to find the sum of first 100 numbers first use the formula n(n+1)/2 where n is the number of terms. in this question 100(100+1)/2=5050 to find the average sum of all observations/no. of observations =5050/100 =50.5

Priya Samidurai
Jun 22, 2017

sum=n(n+1)/2, avg=sum/n

Ogochukwu Ekeocha
Jun 21, 2017

duhh 50.5 find the sum of terms and divide by 100

G Priya
Dec 31, 2016

a1=1; d=1; an=1 ;n=100 sn=n/2[a1+an] sn=100/2[1+100] sn=50[101]/100 {/100 because of the average} sn=50.50

Anurag Pandey
Aug 31, 2016

Sum of first $n$ natural numbers is equal to $\frac{n(n+1)}{2}$ so the average will be equal to

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

So the answer of given question becomes $\frac {100+1}{2} = \boxed{50.5}$

Niloy Debnath
Jun 17, 2016

As 1+..........+10=55.11+............+20=155.21+...........+30=255.Like this, we get (55+155+255+355+455+555+655+755+855+955) which is equals to 5050.And the average is 5050/100=50.50.It's the correct answer.

Positive Integers

13 solutions

0 pending reports