Question

Use Gauss's approach to find a formula for
the sum of the even numbers from 2 to 2n:
2 + 4 + 6 + ... + 2n
Your formula will be an expression involving n.

Answer 1

Answer:

sum = n (n+1)

Step-by-step explanation:

Let S = sum

Gauss's approach

S = 2     + 4        + 6    + ...  + 2n -4  + 2n-2  +  2n

S = 2n + 2n-2 + 2n-4  +  ... +  6         + 4         + 2

add both equations term by term in order

2S = 2n+2  +  2n+2  + 2n+2 ..... (n times

therefore

S = n(n+1)

check:

n = 1,  S = 1(1+1) = 2   [= 2]

n = 2,  S = 2(2+1) = 6   [= 2+4]

...

checks

Answer 2

Gauss's approach was introduced by Carl Friedrich Gauss according to legend, while he was in primary school

The sum of 2 + 4 + 6 +...+ 2·n is n·(n + 1)

The reason the above expression is correct is as follows:

The required numbers to sum are;

Sum 2 to 2·n

The Gauss's approach involves the adding of the first and last and summing the result as follows:

2 × (2 + 4 + ... + 2·n) = (2·n + 2) + (2·n - 2 + 4) + (2·n - 4 + 6)+...+ (2 + 2·n)

Therefore, we have;

2 × (2 + 4 + ... + 2·n)  = n × (2·n + 2)

(2 + 4 + ... + 2·n)  = n × (2·n + 2)/2 = n·(n + 1)

The expression n × (2·n + 2)/2 = (n/2) × (2·n + 2) = (n/2) × (2 + 2·n)

(n/2) × (2 + 2·n) = (number of terms)/2 × (First + Last term)

Number of terms = n

First term = a₁

Last term = a₂

Sum of terms, Sₙ = (n/2) × (a₁ + aₙ) general form by Gauss's approach

Therefore;

The sum of 2 + 4 + 6 +...+ 2·n using the Gauss's approach is therefore;

(2 + 4 + ... + 2·n)  = n·(n + 1)

Learn more about Gauss's approach here:

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