User:Justin Benfield

I have a Bachelor's of Science in Mathematics and have been an avid amateur mathematician for many years (wanting to become a professional eventually).

WIP: For Sum of Sequences of Fifth and Seventh Powers

\(\ds \sum_{i \mathop = 1}^{k + 1} i^5 + \sum_{i \mathop = 1}^{k + 1} i^7\)

\(=\)

\(\ds \sum_{i \mathop = 1}^k i^5 + \sum_{i \mathop = 1}^k i^7 + \paren {k + 1}^5 + \paren {k + 1}^7\)

Definition of Summation

\(\ds \)

\(=\)

\(\ds 2 \paren {\sum_{i \mathop = 1}^k i}^4 + \paren {k + 1}^5 + \paren {k + 1}^7\)

Induction Hypothesis

\(\ds \)

\(=\)

\(\ds 2 \paren {\frac {k \paren {k + 1} } 2}^4 + \paren {k + 1}^5 + \paren {k + 1}^7\)

Closed Form for Triangular Numbers

\(\ds \)

\(=\)

\(\ds 2 \paren {\frac {k \paren {k + 1} } 2}^4 + \paren {k^5 + 5 k^4 + 10 k^3 + 10 k^2 + 5 k + 1} + \paren {k^7 + 7 k^6 + 21 k^5 + 35 k^4 + 35 k^3 + 21 k^2 + 7 k + 1}\)

Binomial Theorem

\(\ds \)

\(=\)

\(\ds 2 \paren {\frac {k \paren {k + 1} } 2}^4 + k^5 + 5 k^4 + 10 k^3 + 10 k^2 + 5 k + 1 + k^7 + 7 k^6 + 21 k^5 + 35 k^4 + 35 k^3 + 21 k^2 + 7 k + 1\)

Associative Law of Addition

\(\ds \)

\(=\)

\(\ds 2 \paren {\frac {k \paren {k + 1} } 2}^4 + k^7 + 7 k^6 + k^5 + 21 k^5 + 5 k^4 + 35 k^4 + 10 k^3 + 35 k^3 + 10 k^2 + 21 k^2 + 5 k + 7 k + 1 + 1\)

Commutative Law of Addition

\(\ds \)

\(=\)

\(\ds 2 \paren {\frac {k \paren {k + 1} } 2}^4 + k^7 + 7 k^6 + \paren {1 + 21} k^5 + \paren {5 + 35} k^4 + \paren {10 + 35} k^3 + \paren {10 + 21} k^2 + \paren {5 + 7} k + \paren {1 + 1}\)

Distributive Laws of Arithmetic

\(\ds \)

\(=\)

\(\ds 2 \paren {\frac {k \paren {k + 1} } 2}^4 + k^7 + 7 k^6 + 22 k^5 + 40 k^4 + 45 k^3 + 31 k^2 + 12 k + 2\)

Arithmetic

\(\ds \)

\(=\)

\(\ds 2 \paren {\frac {\paren {k \paren {k + 1} }^4} {2^4} } + k^7 + 7 k^6 + 22 k^5 + 40 k^4 + 45 k^3 + 31 k^2 + 12 k + 2\)

Quotient of Powers

\(\ds \)

\(=\)

\(\ds \paren {\frac {k^4 \paren {k^4 + 4 k^3 + 6 k^2 + 4 k + 1} } 8} + {k^7 + 7 k^6 + 22 k^5 + 40 k^4 + 45 k^3 + 31 k^2 + 12 k + 2}\)