Sum of All Two-Digit Even Positive Numbers
Identifying and summing all the even numbers within the range of two-digit positive integers can be both an interesting and educational exercise. This article explains the step-by-step process of finding the sum of all two-digit even positive numbers and highlights the mathematical concepts involved.
Understanding Even Numbers
Even numbers are integers that are divisible by 2. In the context of two-digit positive numbers, our focus will be on even numbers ranging from 10 to 98. This range includes numbers like 10, 12, 14, and so on, up to 98. These numbers form an arithmetic sequence, a specific type of sequence where the difference between consecutive terms is constant.
Arithmetic Sequence Definition
An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. For the even numbers within our range, the first term (a) is 10, the last term (l) is 98, and the common difference (d) is 2.
Finding the Number of Terms
To find the total number of terms (n) in this sequence, we use the formula for the n-th term of an arithmetic sequence:
( l a (n - 1) cdot d )
Substituting the known values:
( 98 10 (n - 1) cdot 2 )
Solving for ( n ):
( 98 - 10 (n - 1) cdot 2 )
( 88 (n - 1) cdot 2 )
( n - 1 44 )
( n 45 )
Calculating the Sum of the Sequence
The sum (S) of an arithmetic series can be calculated using the formula:
( S_n frac{n}{2} cdot (a l) )
Substituting the values:
( S_{45} frac{45}{2} cdot (10 98) )
( S_{45} frac{45}{2} cdot 108 )
( S_{45} 45 cdot 54 )
( S_{45} 2430 )
Therefore, the sum of all two-digit even positive numbers is ( 2430 ).
Code Implementation
To verify this sum, we can write a simple program in C :
#include iostreamusing namespace std;int NumSumTwo10() { int SumOfns 0; for (int n 10; n
This C program calculates the sum of all even numbers between 10 and 98, providing the same result of 2430.
Conclusion
Summing all two-digit even positive numbers can be achieved through structured mathematical calculations or simple programming. Whether you're working through a mathematical problem or implementing an algorithm, understanding the underlying concepts can greatly enhance your problem-solving skills.