(2*k + 1)^2 * T(n) + T(k) is a triangular number

 
 

if   T_{n}   is the n-th triangular number then

(2 \, k + 1)^2 \times T_{n} \; + \; T_{k}   is a triangular number
 
 
(2 \, k + 1)^2 \; \times \; n \,(n + 1)/2 \; + \; k \,(k + 1)/2

= \; 1/2 \; (4 \, k^2 \, n^2 + 4 \, k^2 \, n + k^2 + 4 \, k \, n^2 + 4 \, k \, n + k + n^2 + n)

= \; 1/2 \; (2 \, k \, n + k + n) \, (2 \, k \, n + k + n + 1)

Hence,

(2 \, k + 1)^2 \times T_{n} \; + \; T_{k} \; = \; T_{ \,2 \, k \, n + k + n \,}

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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