And I love the words “you just use long division”. ]]>

1 + x + x^2 + x^3 + … + x^(n-1) + x^n is in recursive function terms f(n+1) = f(n) + x^n

————— f(n) ——————

but I can rearrange it as 1 + x(1 +x(1 + x(1 + x(1 + x)))) …to term 5 for example,

which gives

f(n+1) = 1 + xf(n) … in general ….

So eliminating f(n+1) we get

sum = (x^n – 1)/(x – 1)

The rearrangement is the general polynomial for early efficient computing circa 1967

Too tricky for the students I fear !

Sum from 0 to n of “a subscript i” is equal to a(1 – x^n)/(1 – x)

More interesting comment later….