University of Florida/Egm4313/s12.team11.gooding/R3

Problem 3.7

Solved By Kyle Gooding

Problem Statement

Expand the series on both sides of (1),(2) pg. 7-12b to verify these equalities.
(1)

Given

$\sum _{j=2}^5{C}_j*j(j-1)x^{j-2}=\sum _{j=0}^3{C}_{j+2}*(j+2)(j+1)x^j$

(2)
$\sum _{j=1}^5{C}_j*j{x}^{j-1}=\sum _{j=0}^4{C}_{j+1}*(j+1)x^j$

Solutions

Expanding both sides of (1) results in:
$\sum _{j=2}^5{C}_j*j(j-1)x^{j-2}=C_2(2)(2-1)x^0+C_3(3)(3-1)x^1+C_4(4)(4-1)x^2+C_5(5)(5-1)x^3$
$\sum _{j=0}^3{C}_j(j+2)(j+1)x^j=C_{0+2}(0+2)(0+1)x^0+C_{1+2}(1+2)(1+1)x^1+C_{2+2}(2+2)(2+1)x^2+C_{3+2}(3+2)(3+1)x^3$

Simplifying:
$\sum _{j=2}^5{C}_j*j(j-1)x^{j-2}=C_2(2)+C_{3}6x+C_{4}12x^2+C_{5}20x^3$
$\sum _{j=0}^3{C}_j(j+2)(j+1)x^j=C_2(2)+C_{3}6x+C_{4}12x^2+C_{5}20x^3$

 The two sums are equal.

Expanding both sides of (2) results in:
$\sum _{j=1}^5{C}_j*j{x}^{j-1}=C_1(1)x^0+C_2(2)x^1+C_3(3)x^2+C_4(4)x^3+C_5(5)x^4$
$\sum _{j=0}^4{C}_{j+1}*(j+1)x^j=C_{0+1}(0+1)x^0+C_{1+1}(1+1)x^1+C_{1+2}(2+1)x^2+C_{2+2}(3+1)x^3+C_{3+2}(4+1)x^4$

Simplifying:
$\sum _{j=1}^5{C}_j*j{x}^{j-1}=C_1(1)+C_2(2)x^1+C_3(3)x^2+C_4(4)x^3+C_5(5)x^4$
$\sum _{j=1}^5{C}_j*j{x}^{j-1}=C_1(1)+C_2(2)x^1+C_3(3)x^2+C_4(4)x^3+C_5(5)x^4$

 The two sums are equal.

Egm4313.s12.team11.gooding 23:49, 19 February 2012 (UTC)