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?
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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