Sum of the First 50 Natural Numbers: Techniques and Formulas

The sum of the first n natural numbers can be calculated using different methods, including formulas and arithmetic sequences. Here, we will explore how to find the sum of the first 50 natural numbers using the most common and efficient techniques.

Method 1: Using the Arithmetic Sequence Summation Formula

The sum of the first 50 natural numbers can be calculated using the formula for the sum of an arithmetic sequence. The arithmetic sequence summation formula is given by:

Sn {n(a l)}{2}, where n is the number of terms, a is the first term, and l is the last term.

S50  {50 * (1   50)}{2}  {50 * 51}{2}  1275

By substituting the values, we get:

Sum  {50 * 51}{2}  1275

Therefore, the sum of the first 50 natural numbers is 1275.

Method 2: Using the Formula for the Sum of First n Natural Numbers

General Formula for the Sum of First n Natural Numbers

The sum of the first n natural numbers is given by the formula:

S {n(n 1)}{2}.

S50  {50(50   1)}{2}  {50 * 51}{2}  1275

By substituting the values, we get the same result:

Sum  1275

This method is a direct application of the formula and is straightforward to use.

Alternative Methods and Additional Information

Arithmetic Progression (AP) Formula

For an arithmetic progression (AP), the sum of the first n terms is given by:

Sn {n}{2} (2a (n - 1)d), where a is the first term, d is the common difference, and n is the number of terms.

For the first 50 natural numbers:

S50  {50}{2} (2 * 1   (50 - 1) * 1)  25 (2   49)  25 * 50  1250

However, this method is less efficient for this specific problem since the common difference is 1, and the direct formula is simpler.

Geometric Progression (GP) Formula

Note that the sequence of natural numbers is not a geometric progression (GP) as there is no common ratio between consecutive terms. However, the formula for the sum of GP is:

Sn a(rn - 1) / (r - 1), where a is the first term, r is the common ratio, and n is the number of terms.

Since a GP is not applicable here, let us focus on the arithmetic methods.

Common Misconceptions and Additional Calculations

A common misconception arises when using the average method. The average of the first and last term is 25.5, and the sum can be calculated as:

Average  {1   50}{2}  25.5Sum  25.5 * 50  1275

This method also gives the correct result of 1275.

Conclusion

In conclusion, the sum of the first 50 natural numbers is 1275. This can be calculated using the formula for the sum of an arithmetic sequence, the formula for the sum of the first n natural numbers, or the average method. Understanding these methods enhances your problem-solving skills and makes it easier to handle similar problems in the future.