# Quod Erat Demonstrandum

## 2013/06/20

### just-a-q

Filed under: NSS — johnmayhk @ 3:10 下午
Tags:

Just set a so-called senior form core mathematics question:

Q.1 Find the general term $T_n$.

Solution

$T_1=1$

$T_2=1+6$

$T_3=1+6+6\times 2$

$T_4=1+6+6\times 2+6\times 3$

$\dots$

Observe the pattern,

$T_n$

$=1+6+6\times 2+6\times 3+\dots+6\times (n-1)$

$=1+6(1+2+3+\dots +(n-1))$

$=1+3n(n-1)$

Q.2 Observe the following pattern, guess the ongoing patterns and prove it.

$T_{101}=30301$

$T_{1001}=3003001$

$T_{10001}=300030001$

$T_{100001}=30000300001$

$\dots$

$T_{201}=120601$

$T_{2001}=12006001$

$T_{20001}=1200060001$

$T_{200001}=120000600001$

$\dots$

Solution

Not difficult to obtain something like $3a^2(10^{2n})+3a(10^n)+1$

Q.3 Explore more patterns about $T_n$.

Solution

Urm, try something like $T_{66...67}$, $T_{166...67}$ etc.