Sigma notation, also known as summation notation, is a convenient and compact way of writing long su

Question

Sigma notation, also known as summation notation, is a convenient and compact way of writing long sums. It denotes the sum of a sequence of numbers.

The symbol $r$ is the index of summation, or dummy variable.

Important Notes

$S_n$ denotes the sum of the first $n$ terms: \[S_n=u_1+u_2+u_3+\cdots+u_n=\sum_{r=1}^{n}u_r.\]
$S_\infty$ denotes the sum to infinity: \[S_\infty=u_1+u_2+u_3+\cdots=\sum_{r=1}^{\infty}u_r.\]
Recover terms from partial sums using $u_1=S_1$ and $u_n=S_n-S_{n-1}$ for $n\geq 2$. Equivalently, define $S_0=0$.
The total number of terms in $\sum_{r=a}^{b}u_r$ is $b-a+1$.

The following results collect the constant, arithmetic-progression and geometric-progression sums used throughout this unit.

Sequence

Result

Constant term, $u_r=k$

\[\sum_{r=1}^{n}k=\underbrace{k+k+\cdots+k}_{n\text{ terms}}=nk\]

\[\sum_{r=a}^{b}k=\underbrace{k+k+\cdots+k}_{b-a+1\text{ terms}}=(b-a+1)k\]

Arithmetic progression

\[\text{Sum of AP}=\frac{\text{number of terms}}{2}(\text{first term}+\text{last term})\]

\[\sum_{r=1}^{n}r=\underbrace{1+2+\cdots+n}_{n\text{ terms}}=\frac{n}{2}(1+n)\]

\[\sum_{r=a}^{b}r=\underbrace{a+(a+1)+\cdots+b}_{b-a+1\text{ terms}}=\frac{b-a+1}{2}(a+b)\]

Geometric progression, $u_r=a^r$

\[\text{Sum of GP}=\frac{(\text{first term})\bigl(q^{\text{number of terms}}\hspace{0.5em}-1\bigr)}{q-1}\]

\[\sum_{r=1}^{n}a^r=\underbrace{a+a^2+\cdots+a^n}_{n\text{ terms}}=\frac{a(a^n-1)}{a-1}\]

\[\sum_{r=k}^{m}a^r=\underbrace{a^k+a^{k+1}+\cdots+a^m}_{m-k+1\text{ terms}}=\frac{a^k(a^{m-k+1}-1)}{a-1}\]

Accepted Answer

$\sum_{r=3}^{6}(2r-1)=32$, with $4$ terms.

Sigma Notation and Common Summation Results