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Tpy6a [65]
3 years ago
10

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

Mathematics
1 answer:
Gelneren [198K]3 years ago
6 0

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.

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