Please explain Vn method with example and ALGORITHM ..?

Hii

The method of differences is used when we are not given the sum of the series. For example, if we wanted to find the sum, ∑nr=1ur of some series where the terms follow some certain pattern. The aim of the method of differences is to express the rth term, which will be a function of r as the difference of two expressions in r of the same form. What this means is that we write the term ur in the form f(r)−f(r+1) or even f(r+1)−f(r), where f(r) is some function of r. If we can express ur in such a way we can see quite simply that
u1+u2=f(1)−f(2)+f(2)−f(3)=f(1)−f(3)
We can extend this idea further to find
r=1u1=f(1)−f(2)
r=2u2=f(2)−f(3)
r=3u3=f(3)−f(4)
r=n−1un−1=f(n−1)−f(n)
r=nun=f(n)−f(n+1)
Now we can see if we add all of these togetheru1+u2+u3+⋯+un−1+un=f(1)−f(2)+f(2)−f(3)+f(3)−f(4)+⋯+f(n−1)−f(n)+f(n)−f(n+1)=f(1)−f(n+1)


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