Mathematics Advanced: Sequences and Series

• There are 3 types of sequences: arithmetic, geometric, and summing
• Sequences are also known as progressions

Arithmetic Sequences

• In arithmetic sequences, the difference between successive terms is constant:

  Examples:

  $d=T_n-T_{n-1},n\ge 2$

  $T_n-a+(n-1)d\text{ where }a=T_1$

• Suppose 3 numbers, $\text{a, b, }$and $\text{m}$, form an arithmetic sequence when $m-a=b-m$, which can be rearranged as $m=\frac{a+b}{2}$:
  • $m$ is called the arithmetic mean of $a$ and $b$

Geometric Sequences

• Geometric series have a non-zero ratio between successive terms:

Examples:

$\frac{T_n}{T_{n-1}}=r,n\ge 2$

$T_n=a\cdot r^{n-1}\text{ where }a=T_1$

• Three numbers, $\text{a, b, }$and $\text{m}$, form a geometric progression, when $\frac{b}{g}=\frac{g}{a}$, and therefore when $g^2=a\cdot b$
• $g$ is the geometric mean of a, b

Summing Sequences

• Summing sequences occur when an infinite number of terms are added together: $$S_n=T_1+T_2+T_3+\dots +T_n$$
• Summing sequences can be represented using sigma notation:

$$\sum _{k=1}^n{2}^{-n}=T_1+T_2+T_3+\dots +T_n=S_n$$

Sum of an Arithmetic Progression

Let $l=T_n$ be the last term of an AP with $T_1=a$ and difference $d$. It can be therefore determined that:

• $S_n=a+(a+d)+(a+2d)+(a+3d)+\dots +(l-2d)+(l-d)+l$
• $S_n=l+(l-d)+(l-2d)+...+(a+2d)+(a+d)+a$
• Combining the two equations gives $2S_n=n(a+l)$, as there are $n$ terms
• Therefore, $S_n=\frac{1}{2}n(a+l)$
• Since $l=T_n=a+(n-1)d,$ we can derive an alternative formula:
  • $S_n=\frac{1}{2}n(2a+(n-1)d)$
  • $\therefore S_n=\sum _{k=1}^n(a+(k-1)d)$

Sum of a Geometric Progession:

To find the sum of a geometric sequence:

• $S_n=a+ar+a{r}^2+\dots +a{r}^{n-2}+a{r}^{n-1}$
• $r\cdot S_n=ar+a{r}^2+a{r}^3+\dots +a{r}^{n-1}+a{r}^n$

Subtract the second equation from the first:

• $(r-1)S_n=a{r}^n-a$
• $S_n=\frac{a(r^n-1)}{r-1}\text{ when }\mid r\mid <1$

For when $\mid r\mid >1,$ subtract the first equation from the second:

• $(1-r)S_n=a-a{r}^n$
• $S_n=\frac{a(1-r^n)}{1-r}\text{ when }\mid r\mid >1$

Limiting Sum

If $\mid r\mid <1,$ then ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{r}^n=0,}$ therefore:

• ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{T}_n=0}$
• $S_{\mathrm{\infty }}={\displaystyle \underset{n\to \mathrm{\infty }}{lim}{S}_n=\frac{a}{1-r}}$
• Progressions with limiting sums are said to converge on a value
  • For example, $\sum _{n=1}^{\mathrm{\infty }}a{r}^{n-1}\text{ converges to }\frac{a}{1-r}$

Representing Recurring Decimals

If we want to express $1.1\overline{037}$ as a fraction, we can write it as a Geometric Progression $(1.1+0.0037+0.0000037+\dots ):$

• In the brackets, there is a limiting sum: $S_{\mathrm{\infty }}=\frac{0.0037}{1-0.001}$
• $s_{\mathrm{\infty }}=\frac{1}{270}$
• BUT we need to add 1.1 to this sum:
• $\therefore 1.1\overline{037}=\frac{149}{135}$