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) $$ \sum_{j=1}^{5}C_j*jx^{j-1}= \sum_{j=0}^{4}C_{j+1}*(j+1)x^{j}$$

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: $$\sum_{j=2}^{5}C_j*j(j-1)x^{j-2}=C_2(2)+ C_36x+C_412x^2 + C_520x^3 $$ $$\sum_{j=0}^{3}C_j(j+2)(j+1)x^{j}=C_2(2)+ C_36x+C_412x^2 + C_520x^3 $$

The two sums are equal. Expanding both sides of (2) results in: $$ \sum_{j=1}^{5}C_j*jx^{j-1}= C_1(1)x^0+ C_2(2)x^1+C_3(3)x^2+C_4(4)x^3 + C_5(5)x^4 $$ $$ \sum_{j=0}^{4}C_{j+1}*(j+1)x^{j}= C_{0+1}(0+1)x^0+ C_{1+1}(1+1)x^1+C_{1+2}(2+1)x^2+C_{2+2}(3+1)x^3 + C_{3+2}(4+1)x^4 $$

Simplifying: $$ \sum_{j=1}^{5}C_j*jx^{j-1}= C_1(1)+ C_2(2)x^1+C_3(3)x^2+C_4(4)x^3 + C_5(5)x^4 $$ $$ \sum_{j=1}^{5}C_j*jx^{j-1}= C_1(1)+ C_2(2)x^1+C_3(3)x^2+C_4(4)x^3 + C_5(5)x^4 $$ The two sums are equal.

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