Solving Explicity Formula for Sum of i from 1 to n

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In summary, the explicit formula for the sum of i from 1 to n is n(n+1)/2. It is derived using the arithmetic series formula and is significant because it allows for quick and easy calculation of consecutive numbers. It is commonly used in mathematical and scientific calculations and cannot be used for non-consecutive numbers.
  • #1
steveT
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Can anyone help me out with this one?


I need to find an explicit formula for:

n 2
∑ i
i=1

I was already asked to find the explicit polynomial formula for the above equation which is

n·(n + 1)·(2·n + 1)
_________________
6

I'm not sure on what the difference is between the 2 formulas.

Thanks
 
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  • #2
Welcome to PF.

You are correction in your suspicion, the explicit polynomial formula is also the explicit formula.
 

What is the explicit formula for the sum of i from 1 to n?

The explicit formula for the sum of i from 1 to n is n(n+1)/2.

How is the explicit formula derived?

The explicit formula is derived using the arithmetic series formula, Sn = (n/2)(a1+an), where Sn is the sum of n terms, a1 is the first term, and an is the nth term. By substituting 1 for a1 and n for an, the formula simplifies to n(n+1)/2.

What is the significance of the explicit formula for the sum of i from 1 to n?

The explicit formula is significant because it allows us to quickly and easily calculate the sum of any consecutive numbers from 1 to n without having to add each individual term. This is particularly useful in mathematical and scientific calculations.

How is the explicit formula used in real-world applications?

The explicit formula for the sum of i from 1 to n is commonly used in fields such as physics, engineering, and finance to calculate quantities such as the total distance traveled, the total force exerted, or the total cost of a project.

Can the explicit formula be used for non-consecutive numbers?

No, the explicit formula for the sum of i from 1 to n is only applicable for consecutive numbers starting from 1. For non-consecutive numbers, a different formula, such as the arithmetic series formula, must be used.

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