Finding integers

Question

What is the sum of integers from 16 to 37?

Answer 1

You could add the integers (by hand, using a calculator or computer) or you could use Gauss' famous formula for the sum of the numbers from 1 to n:

        Sum(from 1 to n) = n*(n+1)/2

 

    Note (example):         1  +  2  +  3  +  4  =  S

                                      +  4  +  3  +  2  +  1  =  S

                                         -----------------------------

                                          5  +  5  +  5  +  5   +  2S

        

                        Four Fives make twice the desired sum.

 

"sum of integers from 16 to 37"     means

    "sum of the integers from 1 to 37"  minus  "sum of the integers from 1 to 15"

 

      Sum =  37*38/2  -  15*16/2

      Sum =  703 - 120

      Sum =  583

Answer 2

A slightly different way to look at this question.

16,17,18,...,37 is an arithmetic progression with a₁=16, d=1 and a_n=37 and n=37-16+1=22. 

Sum of n first elements od arithmetic progression is S_n=(n/2)(a₁+a_n). So,

S₂₀=(a₁+a₂₀)*(22/2)=11*(16+37)=11*53=583.

 

Answer 3

Understanding the question is always key in solving problems.

 

Integers are simply whole numbers.

Sum is to add.

 

This asks that you find the total of numbers from 16 to 37, which list all numbers in this range and add.

 

16, 17, 18 and so on until 37.

 

Add all numbers and that is your sum of these integers.