Triangular numbers| T(2)*T(4)*T(6)*…*T(2n) = k*T(1)*T(3)*T(5)*…*T(2n-1)

 
 

T_2 \cdot T_4 \cdot T_6 \; ... \; T_{2n} \; = \; k \cdot ( \,T_1 \cdot T_3 \cdot T_5 \; ... \; T_{2n-1} \,)

Show that   k   is always an integer

 

For example,
 

T_2 \; = \; 3 \cdot T_1

T_2 \cdot T_4 \; = \; 5 \cdot ( \,T_1 \cdot T_3 \,)

T_2 \cdot T_4 \cdot T_6 \; = \; 7 \cdot ( \,T_1 \cdot T_3 \cdot T_5 \,)

T_2 \cdot T_4 \cdot T_6 \cdot T_8 \; = \; 9 \cdot ( \,T_1 \cdot T_3 \cdot T_5 \cdot T_7 \,)

 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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