Solved By Kyle Gooding
Expand the series on both sides of (1),(2) pg. 7-12b to verify these equalities.
(1)
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1469a35296f544c13e37b4185fd6f8e21705a66d)
(2)
![{\displaystyle \sum _{j=1}^{5}C_{j}*jx^{j-1}=\sum _{j=0}^{4}C_{j+1}*(j+1)x^{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54af7703ca43a36d55e452de43914530e8f51511)
Expanding both sides of (1) results in:
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97afe15b8fdcde693d1b81c71ac7f55fe08c1d66)
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5be219e386f13fe59334f319a8da14c55f5dbbb9)
Simplifying:
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/266b7dedea50f13ba9b610f8c26fab6e5c5bce8d)
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0886b23775a48cf504a30be80ca3ad919ad1ff1)
The two sums are equal.
Expanding both sides of (2) results in:
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31e391c28f7981d0edd72aec565c7bae9a5cf8a2)
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7aae71ae04e22c932d7cc2644b805b06dbdafecf)
Simplifying:
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef343b1d734f21d0c899756d982002498907827a)
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef343b1d734f21d0c899756d982002498907827a)
The two sums are equal.
Egm4313.s12.team11.gooding 23:49, 19 February 2012 (UTC)