My maths teacher showed me something on how to calculate sums. Let's take an example:
n∑k=1k(k+1)=n∑k=1k2+n∑k=1k=n(n+1)(2n+1)6+n(n+1)2=n(n+1)(n+2)3
This was an easy one, but I just can't understan how to solve such sums:
n∑k=1(k−1)k(k+1)
n∑k=11(3n−2)(3n+1)
Could anybody help me, please?
I want to understand the idea of solving sums like these, so please, do not be very specific, but help me giving these and maybe some other examples.
Answer
HINT:
The second example is orthogonal to the first, hence a different answer
3(3n+1)(3n−2)=(3n+1)−(3n−2)(3n+1)(3n−2)=13n−2−13n+1
Set a few values of n=1,2,3,⋯,n−2,n−1,n to recognize the Telescoping Series
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