MHB Fast Calculation [sum] problem

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To solve the fast calculation problem involving the sum of three questions, three summation formulas are provided: the sum of the first n natural numbers, the sum of the squares of the first n natural numbers, and the sum of the cubes of the first n natural numbers. Each formula requires identifying the value of n to compute the respective sums. Users are encouraged to apply these formulas to their specific problems for accurate results. The discussion emphasizes the importance of correctly determining n for effective calculations.
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hello all...

would you help me out, how to sum 3 questions problem..

please, see my picture attachment..

thanks for your helping...

susanto3311
 

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There are 3 formulas you can use here:

[box=blue]
Summation Formulas

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

$$\sum_{k=1}^n\left(k^2\right)=\frac{n(n+1)(2n+1)}{6}\tag{2}$$

$$\sum_{k=1}^n\left(k^3\right)=\left(\frac{n(n+1)}{2}\right)^2\tag{3}$$[/box]

Can you identify $n$ in each case and find the sums?
 
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