Problem 94 - Algebra and Trigonometry

Question

Show that $$ 1+2+\cdots+(n-1)+n=\frac{n(n+1)}{2} $$ [Hint: Let $$ \begin{array}{l} S=1+2+\cdots+(n-1)+n \\ S=n+(n-1)+(n-2)+\cdots+1 \end{array} $$Add these equations. Then $$ 2 S=[1+n]+[2+(n-1)]+\cdots+[n+1] $$

Step-by-Step Solution

Answer

The sum of the first n natural numbers is \( S = \frac{n(n+1)}{2} \).

Step 1: Define the Summation

Consider the given series sum as S. Let the sum of the series be represented by the expression: \[ S = 1 + 2 + 3 + \ldots + (n-1) + n \]

Step 2: Reverse the Series

Write the series in reverse order: \[ S = n + (n-1) + (n-2) + \ldots + 2 + 1 \]

Step 3: Add the Two Representations

Add the original series and the reversed series: \[ 2S = (1 + n) + (2 + (n-1)) + (3 + (n-2)) + \ldots + ((n-1) + 2) + (n + 1) \]

Step 4: Simplify the Sum

Each term in the pair (1 + n), (2 + (n-1)), etc., sums to (n+1). Since there are 'n' terms, we can write: \[ 2S = n \cdot (n+1) \]

Step 5: Solve for S

Divide both sides of the equation by 2 to isolate S: \[ S = \frac{n(n+1)}{2} \]

Step 6: Conclusion

Thus, we have proved the given expression.

Key Concepts

arithmetic series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is always constant. This difference is known as the 'common difference.' For example, in the series 2, 5, 8, 11, ..., the common difference is 3 because each term increases by 3.

The formula to find the sum of the first n terms of an arithmetic series is key. For an arithmetic series starting with the first term a_1 and a common difference d, the nth term a_n is given by:
\(a_n = a_1 + (n-1) \times d\)

The sum S_n of the first n terms of the arithmetic series is:
\[S_n = \frac{n}{2} \times (a_1 + a_n)\]

• If the common difference is 1, the arithmetic series becomes the sum of the first n natural numbers.
• The sum of the first n natural numbers can be simplified as we did in the exercise.

mathematical proof

Mathematical proof is a logical argument demonstrating the truth of a statement. It’s constructed using deductive reasoning and follows a sequence of statements that lead to a conclusion.

The proof we used for the sum of the first n natural numbers employs a clever technique involving the reverse series. By adding the original series to its reverse, we show that each pair of terms sums to a constant value (n+1). Let's break this down further:

• First, represent the sum of the series as S.
• Then reverse the series and represent it again as S.
• By adding both representations, each term sums to (n+1).

By observing that there are n such pairs, we deduce:
\[2S = n \times (n + 1)\]
Finally, dividing by 2 isolates S and proves the formula:
\[S = \frac{n(n+1)}{2}\]

summation formula

A summation formula helps in quickly calculating the sum of a series without manually adding every term. For the first n natural numbers, this formula is \[S = \frac{n(n+1)}{2}\]

Here's how it works:

• The numerator n(n+1) represents the product of n and the next number (n+1).
• The denominator 2 accounts for the pairing of terms as we observed in the proof.

To understand why this formula works, remember that by pairing the numbers in the series (1 with n, 2 with n-1, etc.), each pair sums to the same value (n+1). We then account for all pairs by multiplying by n and dividing by 2.

This summation formula is widely used in series calculations and has practical applications in mathematics, computer science, and various fields involving numerical computations.