Properties of Summation and Standard Results

Properties of Summation: (i) $\sum_{r=1}^n kt_r=k\sum_{r=1}^n t_r$ for a nonzero constant $k$; (ii) $\sum_{r=1}^n(a_r+b_r)=\sum_{r=1}^na_r+\sum_{r=1}^nb_r$; (iii) $\sum_{r=1}^n 1=n$; (iv) $\sum_{r=1}^{n} k = kn$ for a nonzero constant $k$ (a restatement of (i) and (iii) together).

Result 1: the sum of the first $n$ natural numbers is $\sum_{r=1}^{n}r=\dfrac{n(n+1)}2$.

Result 2: the sum of the squares of the first $n$ natural numbers is $\sum_{r=1}^{n}r^2=\dfrac{n(n+1)(2n+1)}6$.

Result 3: the sum of the cubes of the first $n$ natural numbers is $\sum_{r=1}^{n}r^3=\left[\dfrac{n(n+1)}2\right]^2$ (these three results can be formally proved using Mathematical Induction, covered later in the book).

Worked Example 1: evaluate $\sum_{r=1}^{n}(8r-7)$. $=8\sum r-7\sum1=8\cdot\dfrac{n(n+1)}2-7n=4n^2+4n-7n=4n^2-3n$.

Worked Example 2: find $3^2+4^2+5^2+\cdots+29^2$. $=\sum_{r=1}^{29}r^2-\sum_{r=1}^{2}r^2=\dfrac{29\cdot30\cdot59}6-\dfrac{2\cdot3\cdot5}6=(29\times5\times59)-5=5(29\times59-1)=5(1710)=8550$.

Worked Example 3: find $100^2-99^2+98^2-97^2+\cdots+2^2-1^2$. Grouping the even-indexed and odd-indexed squares separately: $=\sum_{r=1}^{50}(2r)^2-\sum_{r=1}^{50}(2r-1)^2=\sum_{r=1}^{50}(4r-1)=4\cdot\dfrac{50\cdot51}2-50=5100-50=5050$. …