Q.In a certain town, a viral disease caused severe health hazards upon its people disturbing their normal life. It was found that on each day, the virus which caused the disease spread in Geometric Progression. The amount of infectious virus particles gets doubled each day, being $5$ particles on the first day. Find the day when the infectious virus particles just grow over $1{,}50{,}000$ units.

A sequence is an ordered list of numbers; formally, a finite sequence on $n$ terms is a function $f:\{1,2,\ldots,n\}\to\mathbb R$ with $f(k)=a_k$, while an infinite sequence is a function on all of $\mathbb N$. Terms may repeat (unlike the elements of a set).

Arithmetic progression (AP): $a,\,a+d,\,a+2d,\ldots$, where $d$ is the common difference. The $n^{th}$ term is

Every term (after the first) is the arithmetic mean of its neighbours: $a_k=\dfrac{a_{k-1}+a_{k+1}}2$ (Result 5.1) — equivalently, $T_{m+n}+T_{m-n}=2T_m$ for any valid $m,n$.

Geometric progression (GP): $a,\,ar,\,ar^2,\ldots$ ($a\ne0,r\ne0$), where $r$ is the common ratio. The $n^{th}$ term is

Every term is the geometric mean of its neighbours: $a_k=\sqrt{a_{k-1}a_{k+1}}$ (Result 5.2), so $t_{n-k},t_n,t_{n+k}$ is again a GP for any $k$. Taking logs of a GP with $r>0$ turns it into an AP with common difference $\log r$.

Arithmetico-geometric progression (AGP): term-by-term product of an AP and a GP, $a,(a+d)r,(a+2d)r^2,\ldots$, with $n^{th}$ term $T_n=(a+(n-1)d)r^{n-1}$. Setting $r=1$ recovers an AP; setting $d=0$ recovers a GP — AP and GP are special cases of AGP.

Harmonic progression (HP): a sequence whose reciprocals form an AP: $h_1,h_2,\ldots$ is an HP exactly when $\dfrac1{h_1},\dfrac1{h_2},\ldots$ is an AP, giving the general HP form $\dfrac1a,\dfrac1{a+d},\dfrac1{a+2d},\ldots$ (provided no denominator vanishes). If $a,b,c$ are in HP then $b=\dfrac{2ac}{a+c}$. …