what is the formula of adding all number from 1 to nth number?

Answer 1

It's easy to derive. Take 1+2+3+4+5+6:

This is the same as (1+6) + (2+5) + (3+4). All of these values are the same, 7. So this is 3*7, or (half the number) * (one more than the number).

Try it for up to 10; similarly we get (1+10) + (2+9) + ...

It even works for odd numbers. For 1 to 5, we have (1+5) + (2+4) + 3, or 6 + 6 + 3, or (2.5) * 6. This is again (half the number) * (1 + the number).

So the formula is n(n+1)/2.

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We can prove this formula with induction.

Obviously it's true for n=1: 1(2)/2 = 1.

So suppose it's true for some n. Then if you add n+1 to both sides,
(sum of 1 to n) + (n+1) = n(n+1)/2 + (n+1)
(sum of 1 to n+1) = (n+1) [n/2 + 1] = (n+1) (n+2)/2
This is the same formula, except for n+1 instead of n.

So we know that it's true for 1, and that if it's true for 1 it's true for 2, and if it's true for 2 it's true for 3, etc. And therefore it's true for all natural numbers.

Answer 2

Everyone has the answer correct so far, and this is nice.

I'd just like to note something about Geofft's inductive proof.

After we show that n(n+1)/2 is true for n=1, then we want to show that it is true for n=1+1, and also for n=1+2, and for any number n, and thus for any number n+1, n+2, etc.

Instead of adding, you can think of it as replacing, or plugging in, different values to show that it works for all of them.

Geofft obviously clearly understands this, and I think he was trying to simplify the language of inductive proofs when he used the term add. I apologize in advance if this upsetting to anyone, I just wanted to clarify the processes.

So, when we run the equation the second time, we are not adding (n+1) to n(n+1)/2: we are substituting the term n+1 for the term n where ever it occurs.

We are trying to show that if something is true for 1, and also true for 2, and also true for 3, then it is also true for any n which is a member of the natural numbers (1, 2, 3, ..., n, n+1, n+2, ...).

So what Geofft was showing was that since the equation is true for 1, and true for values of n, and n+1, then it must logically be true for all values which are Natural numbers.

This is quite a beautiful equation, and a story is told of Gauss, the French Mathematician who, as a boy, was given the task of adding the numbers 1 to 100, so his teacher could get some quiet time. Gauss was done so quickly, and had the correct answer, that the teacher supposedly punished him.

It is known that this story is not true, but so many stories were told of Gauss as a child genius that this one was believed by mathematicians for many years.

Answer 3

n(n+1)/2

Answer 4

When I have need to determine this question I use a simple formula that I devised for myself.Suppose you had a very large triangular frame(equilateral)Into an angle you place a ball,underneath you place two more balls.Now you have a small triangle of two balls each side.Common sense tells you there's 3 balls in triangle,but you're going to prove it.half of 2 is 1
1times2=2plus multiplier1total=3 Correct answer is 3
Now add 3 more balls under the 2 Again common sense says you've added 3 balls to the triangle,so now there's 6 Again we'll prove it You now have a triangle of 3 balls per side
Half of 3 is 1.5 plus .5=2... 2times3 =6 correct answer
So this simple formula works with any number.If the bottom row is an odd number half it and add a half.Multiply the bottom row with this number and this is the answer.When the bottom row is an even number half it multiply bottom row with the number,then add the multiplier to your answer this is the correct number
Suppose there's nine in the bottom row Half of 9 is4 and a half,plus half =5..5 times 9 =45 correct answer Now if you add 10 more to bottom row Half of 10=5...5 times10=50 plus multiplier5...50plus 5=55 correct answer This simple formule works with any number even if the bottom row is 0000s

Answer 5

n(n+1)/2

Answer 6

n*(n+1) / 2

Answer 7

It is the sum of natural numbers from 1 to n.

sum = n(n+1)/2

To find the sum of 10 natural numbers, i.e., from 1 to 10:

sum = 10(10+1)/2 = 10*11/2 = 110/2 = 55

Answer 8

n(n+1)/2

Answer 9

I think you will have a very, very hard time finding a closed formula for this sequence. Since this is work-related, I'm sure you didn't just get this problem to solve. Sometimes it helps if you can provide more background on what you really need, there could be an alternative way to formulate a solution. Perhaps you could outline your actual problem and show how you came to this sequence as the solution. Alternatively, this problem may be only part of your actual problem and if you describe the whole problem there might be another way.

Answer 10

there is a generic formula available..
sum of first n number in a series is
n(n+1)/2