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:

johnmayhk-hexgonal-numbers

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.

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