Adv Alegbra HW Solutions 5

Adv Alegbra HW Solutions 5 - 5 (ii) Prove that Ramanujans...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
5 (ii) Prove that Ramanujan’s statement is false. Solution. One must pay attention to hypotheses. Consider a 3 + b 3 if b is negative: 728 = 12 3 + ( 10 3 ) = 9 3 + ( 1 ) 3 . 1.11 Derive the formula for n i = 1 i by computing the area ( n + 1 ) 2 of a square with sides of length n + 1 using Figure 1.1. Solution. Compute the area A of the square in two ways. On the one hand, A = ( n + 1 ) 2 . On the other hand, A =| D |+ 2 | S | , where D is the diagonal and S is the “staircase.” Therefore, | S |= 1 2 h ( n + 1 ) 2 ( n + 1 ) i = 1 2 n ( n + 1 ). But | S | is the sum we are seeking. 1 2 3 4 51 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Figure 1.1 1 + 2 +···+ n = 1 2 ( n 2 + n ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Figure 1.2 1 + 2 n = 1 2 n ( n + 1 ) 1.12 (i) Derive the formula for n i = 1 i by computing the area
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

Ask a homework question - tutors are online