Question

In this problem we investigate the relationship between a function on integers and the “de- rived function" of its differences. For instance, let f(n) = n2 and define the differences by d(n) = f(n) – f(n − 1). Then d(n) = n2 – (n − 1)2 = n2 – (n– 2n + 1) = 2n – 1. While 2n – 1 is not the derivative of n?, it is related.

Provide a detailed and rigorous proof

Answer 1

2 q Let. Pof (n) z n(n+1)(2n+1) n(n+1) p+ 6 nr +S. 9 + Nowe, f(0) 28. So, the formula is true for noo Let it be true for myo flm) = m(m+1) (2m+1) mim+1) pt q+ mits, mzo 2 Then dlm+1) a flm +1) -f(m) i. e. flm+1) flm) + d(m+1) mim+l) (Rm+1) 6. ie. 6 2 2 2 mm +1) pt q+miast +Plm+1)2 + 9(m+1) + ve mim+1)(2m+1) 6 +(m+347p + [mim+)(m+1)]q +(M11) M + S (m+1) m(2m+1) t = [() m+1} ] p + ferma){+3] 6 + (m+ilor +5.

(+1)&2m² + m +6m+6? z 6 q + (m+ 1)37+ 8 2 pa (m+1)(m+2) [tron) Elmets} ]p» [m$mp3}]q« Imaxes * [ms{ am ramte}] P +(m+1) (m+2) 2m²+3m +um+62 [lm+) { nzurones? [(mi) { (2m+3}{m+2}}]p + {mt){# 2)q+ (mm2173 (m+) (m+j+1)(al(+1)+1) 9+ (m+1)778 9 +(M+1) 5148 2 2 6 q (m+1)(m +1 + + 2 p+ 2 6 Hence P(m+1) is true if Plm) is true , & myo Hence P(n) is true. fin) nin+1)(2n +1) p + n(n+1) 97 101 +3 i.e. 2 2 6

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