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13 Let f(r) = r(r + 1)(r + 2). Show that f(r) – f(r – 1) = 3r(r + 1). Hence show that r(r+ 1) =n(n+ 1)(n+ 2). r=1 Using the standard result for r, deduce that r =n(n+ 1)(2n + 1) %3D r=1 r=1 Find the sum of the series 12 + 2x 22 + 32² + 2x42 + 52 + 2×6² + ...+ 2(n – 1)² +n² , where n is odd.

13 Let f(r) = r(r + 1)(r + 2). Show that f(r) – f(r – 1) = 3r(r + 1). Hence show that r(r+ 1) =n(n+ 1)(n+ 2). r=1 Using the standard result for r, deduce that r =n(n+ 1)(2n + 1) %3D r=1 r=1 Find the sum of the series 12 + 2x 22 + 32² + 2x42 + 52 + 2×6² + ...+ 2(n – 1)² +n² , where n is odd.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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13 Let f(r) = r(r + 1)(r + 2). Show that f(r) – f(r – 1) = 3r(r + 1). Hence show that r(r+ 1) =n(n+ 1)(n+ 2). r=1 Using the standard result for r, deduce that r =n(n+ 1)(2n + 1) %3D r=1 r=1 Find the sum of the series 12 + 2x 22 + 32² + 2x42 + 52 + 2×6² + ...+ 2(n – 1)² +n² , where n is odd.
Transcribed Image Text:13 Let f(r) = r(r + 1)(r + 2). Show that f(r) – f(r – 1) = 3r(r + 1). Hence show that r(r+ 1) =n(n+ 1)(n+ 2). r=1 Using the standard result for r, deduce that r =n(n+ 1)(2n + 1) %3D r=1 r=1 Find the sum of the series 12 + 2x 22 + 32² + 2x42 + 52 + 2×6² + ...+ 2(n – 1)² +n² , where n is odd.
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