Question

How do you add all even numbers that are less than 123?

Answer 1

Answer:

$61(61+1)=3782$

The formula for the sum of the first n positive evens is:

$n(n+1)$. Your question is equivalent to what is the sum of the first 61 even positive integers.

Step-by-step explanation:

122 is the highest even less than 123.

If we solve 2n=122 we can find what number term 122 is in the sequence of positive even numbers.

2n=122

Divide both sides by 2:

n=61

So 122 is the 61st positive even integer. This means we are adding 61 positive even integers.

2(1)+2(2)+2(3)+2(4)+2(5)+2(6)+..........+2(61)

Factor out out 2:

2(1+2+3+4+5+6+...+61)

We could write in summation notation if you prefer:

$2\sum_{k=1}^{61}k$

There is a formula for computing the following:

$\sum_{k=1}^{n}k=\frac{n(n+1)}{2}$

So we have the following:

$2\frac{61(61+1)}{2}$

$61(61+1)$

$61(62)$

$3782$

So if you wanted to know the sum of the first n even numbers it is:

$n(n+1)$.

Examples:

The sum of the first 4 positive even numbers:

$2+4+6+8=20$

Now let's put our formula to the test:

$4(4+1)$

$4(5)$

$20$

The sum of the first 10 positive even numbers:

$2+4+6+8+10+12+14+16+18+20=110$

Now let's put out formula to the test again:

$10(10+1)$

$10(11)$

$110$

So as you can see the formula works.

Let me know if you want me to actually prove with mathematical induction.