What is the formula for adding a sequence of consecutive numbers?

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The discussion revolves around finding the sum of a sequence of consecutive numbers, specifically from 1 to 10,000,000. A pattern is noted where the sum of the first and last numbers in a sequence remains consistent, which leads to a method of pairing numbers to simplify the addition. Participants suggest deriving a formula for the sum of the first n numbers, emphasizing the need for independent attempts and visualization of patterns. The conversation encourages exploring the formula for both even and odd n, highlighting the importance of understanding the underlying mathematics rather than seeking direct answers. Overall, the thread focuses on mathematical reasoning and pattern recognition in summing sequences.
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Homework Statement

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hey hello every one I want to ask a question that is add 1+2+3+4.....10000000. I think its very easy. for example let us add first 5 numbers 1+2+3+4+5 now we can find a pattern that adding first and last digit will be always same. for example 5+1=4+2 and similarly but it is a odd number that we can Ind easily by simple maths. so here we can multiply 6(addition of first and last number)×2(number of pairs) +3 because it is odd number
 
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shina said:

Homework Statement

Homework Equations

The Attempt at a Solution


hey hello every one I want to ask a question that is add 1+2+3+4.....10000000. I think its very easy. for example let us add first 5 numbers 1+2+3+4+5 now we can find a pattern that adding first and last digit will be always same. for example 5+1=4+2 and similarly but it is a odd number that we can Ind easily by simple maths. so here we can multiply 6(addition of first and last number)×2(number of pairs) +3 because it is odd
number


I know I should write question separately but I couldn't
 
So what is your question? Is it simply "how do you add 1 + 2 + 3 + ... + n" ? If so, you need to show some attempt to do this yourself. HINT: it is trivially easy.
 
shina said:

Homework Statement

Homework Equations

The Attempt at a Solution


hey hello every one I want to ask a question that is add 1+2+3+4.....10000000. I think its very easy. for example let us add first 5 numbers 1+2+3+4+5 now we can find a pattern that adding first and last digit will be always same. for example 5+1=4+2 and similarly but it is a odd number that we can Ind easily by simple maths. so here we can multiply 6(addition of first and last number)×2(number of pairs) +3 because it is odd number
The pairing of the greatest with the smallest number and working inwards looks fine.
Can you derive a formula for it, say 1+2+...+n for even n?
And for odd n, what happens if you apply your formula then on n-1 and add n separately?
 
fresh_42 said:
Can you derive a formula for it, say 1+2+...+n for even n?
Yes. For ANY n. Try it. We don't spoon feed answers here, you have to show some work.
 
You may try to guess the formula by visualising the pattern. For example, draw one dot in the first line, two dots in the second, so on and so forth. Can you observe any pattern of the total number of dots?
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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