Progressions

Arithmetic progressions

Definition: An arithmetic progression is a sequence of numbers ${{a}_{1}},{{a}_{2}},{{a}_{3}},...,{{a}_{n}},...$ in which each term, starting from , is obtained from the previous one by adding a constant number called the common difference of the progression. It is written $\underset{\scriptscriptstyle\cdot}{\dot{-}}{{a}_{1}},{{a}_{2}},{{a}_{3}},...,{{a}_{n}},...$

If ${{a}_{1}}$ is the first term, the $n$-th term (the general term), $r$ the common difference, $n$ the number of terms and the sum of the $n$ terms, then we have:

${{a}_{n}}={{a}_{n-1}}+r, n\ge 2$ (by definition)

${{a}_{n}}={{a}_{1}}+(n-1)r, n\ge 2$

$r={{a}_{n}}-{{a}_{n-1}}, n\ge 2$

${{S}_{n}}={{a}_{1}}+{{a}_{2}}+{{a}_{3}}+...+{{a}_{n}}$, ${{S}_{n}}=\frac{({{a}_{1}}+{{a}_{n}})\cdot n}{n}$,

Terms equidistant from the ends

In an arithmetic progression the sum of two terms equidistant from the ends is equal to the sum of the end terms: .

If the number of terms is odd, then there is a middle term ${{a}_{m+1}}$, such that $2{{a}_{m+1}}={{a}_{1}}+{{a}_{2m+1}}$.

The necessary and sufficient condition for three terms $a,b,c,$, taken in that order, to form an arithmetic progression is that .

$\displaystyle \sum\limits_{k=1}^{n}{2k}=n(n+1),n\in \mathbb{N}$ $\displaystyle \sum\limits_{k=1}^{n}{(2k-1)}={{n}^{2}},n\in \mathbb{N}$

Geometric progressions

Definition: A geometric progression is a sequence of numbers ${{a}_{1}},{{a}_{2}},{{a}_{3}},...,{{a}_{n}},...$ in which each term, starting from , is obtained from the previous one by multiplying it by one and the same , called the common ratio. It is written $\underset{..}{\overset{..}{\mathop -}}\, {{a}_{1}},{{a}_{2}},{{a}_{3}},...,{{a}_{n}},...$

If ${{a}_{1}}$ is the first term, the $n$-th term (the general term), $q$ the common ratio, $n$ the number of terms and the sum of the $n$ terms, then we have:

${{a}_{n}}=q\cdot {{a}_{n-1}}, n\ge 2$ (by definition)

${{a}_{n}}={{a}_{1}}\cdot {{q}^{n-1}}, n\ge 2$, $q=\frac{{{a}_{n}}}{{{a}_{n-1}}}$

${{S}_{n}}={{a}_{1}}+{{a}_{2}}+{{a}_{3}}+...+{{a}_{n}}$, ${{S}_{n}}={{a}_{1}}\frac{{{q}^{n}}-1}{q-1}$,

${{S}_{n}}=n\cdot {{a}_{1}},\text{ dac }\!\!\breve{\mathrm{a}}\!\!\text{ }q=1$

Terms equidistant from the ends

In a geometric progression the product of two terms equidistant from the ends is equal to the product of the end terms: ${{a}_{k}}\cdot {{a}_{n-k+1}}={{a}_{1}}\cdot {{a}_{n}}$.

If the number of terms is odd, then there is a middle term ${{a}_{m+1}}$, such that ${{a}^{2}}_{m+1}={{a}_{1}}\cdot {{a}_{2m+1}}$.

The necessary and sufficient condition for three terms $a,b,c,$, taken in that order, to form a geometric progression is that ${{b}^{2}}=a\cdot c$.

Note. If the geometric progression is decreasing $(\left| q \right|<1)$ and the number of terms is infinite, then

${{S}_{\infty }}=\frac{{{a}_{1}}}{q-1}$.