Question

a. Use Gauss's approach to find the following sum: $$ 2+4+6+\ldots+100 $$ b. Use Gauss's approach to find a formula for the sum of the even numbers from 2 to \(2 n\) : $$ 2+4+6+\ldots+2 n $$ Your formula will be an expression involving \(n\).

Short answer

a. The sum is 2550. b. The formula is \(n(n + 1)\).

Step-by-step solution

Understand the Sequence

The sequence we have is 2, 4, 6, ..., 100. This is an arithmetic sequence where the first term \(a = 2\) and the common difference \(d = 2\).

Find the Number of Terms

To find the number of terms in the sequence, use the formula for the last term of an arithmetic sequence: \(a_n = a + (n-1) \cdot d\). Set \(a_n = 100\), \(a = 2\), \(d = 2\) and solve for \(n\):\[100 = 2 + (n-1) \cdot 2\]\[98 = 2(n-1)\]\[49 = n - 1\]\[n = 50\]Therefore, there are 50 terms.

Apply Gauss's Formula for the Sum

Gauss's formula for the sum of an arithmetic series is \(S_n = \frac{n}{2} (a + a_n)\). Substitute \(n = 50\), \(a = 2\), and \(a_n = 100\):\[S_{50} = \frac{50}{2} (2 + 100)\]\[S_{50} = 25 \cdot 102\]\[S_{50} = 2550\]

Generalize for Any n

Now, to find the sum for \(2, 4, 6, ..., 2n\), use Gauss's approach.The last term \(a_n = 2n\), and the number of terms \(n = \frac{2n}{2} = n\).The formula for the sum is:\[S = \frac{n}{2} (2 + 2n)\]\[S = \frac{n}{2} \cdot 2 (1 + n)\]\[S = n(n + 1)\]

Final Step: Solution Verification

Verify the formula with a specific example. If \(n=50\), the sum should be \(50 \cdot 51 = 2550\), which matches the calculation for part (a).

Key concepts

Gauss's Formula

Gauss's Formula is a fascinating and efficient way to sum arithmetic sequences. This method was famously used by Carl Friedrich Gauss, a brilliant mathematician, when he was just a child. He used it to effortlessly find the sum of an arithmetic sequence, which saved time and effort compared to traditional methods.
Gauss's Formula for the sum of the first \( n \) terms of an arithmetic sequence is:

• \( S_n = \frac{n}{2}(a + a_n) \)

Here, \( S_n \) is the sum of the sequence, \( n \) is the number of terms, \( a \) is the first term, and \( a_n \) is the last term. By pairing terms from both ends of the sequence, Gauss could see that these pairs all had the same sum. This insight allows the formula to be very efficient for calculating the total sum.
This approach is particularly helpful for large sequences where calculating each term individually would be cumbersome.

Sum of Even Numbers

The sum of even numbers forms a special kind of arithmetic series that is both simple and systematic. When you need to find the sum of even numbers, you're actually summing over an arithmetic sequence where the difference between consecutive terms is constant at 2.
This series can be represented as:

• \( 2, 4, 6, \ldots, 2n \)

For example, if you want to sum all even numbers from 2 to 100, you calculate it as an arithmetic sequence. The first term \( a = 2 \), and the common difference \( d = 2 \), with the last term being \( 2n = 100 \) in this case. Using the formula for the sum of even numbers, which leverages Gauss's Formula, can make this a breeze.
Recognizing these series patterns helps streamline calculations and allows the use of precise formulas that directly leverage arithmetic properties.

Arithmetic Sequence

An Arithmetic Sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference, \( d \), to the previous term. It follows this general form:

• \( a, a+d, a+2d, a+3d, \ldots \)

Here, \( a \) represents the first term, and \( d \) is the common difference. To find any term in an arithmetic sequence, you can use the formula:

• \( a_n = a + (n-1)d \)

This formula helps identify any specific term in the sequence without listing all prior terms. Identifying the structure of arithmetic sequences makes it easier to handle various problems, especially when working with more extensive sequences in exercises like finding the sum of even numbers using Gauss's method.

Sequence Formula

Understanding the Sequence Formula is key to solving problems involving sums of sequences. Arithmetic sequences and formulas are straightforward once you know the basic concepts involved. The Sequence Formula, especially in arithmetic series, helps determine characteristics such as the sum, the number of terms, and the nth term.
For the sum of the first \( n \) even numbers, the formula simplifies to:

• \( S = n(n + 1) \)

This formula arises from manipulating and generalizing the standard sum formula \( S_n = \frac{n}{2} (a + a_n) \).
By understanding this, you can generalize the approach to any sequence of even numbers, making it straightforward to compute large sums without performing repetitive addition. Using this formula, the calculations become more manageable and efficient, underscoring the power of mathematical patterns and formulas in tackling otherwise complex numerical problems.