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Triangular numbers are defined as Tn:=1+2+...+n for n>= 1. Find a simple formula for Tn+Tn-1 and prove it.

I know Tn=n(n+1)/2 so would Tn-1=n(n-1)/2?

And then would I prove it by just adding Tn+Tn-1?

Antonio Vargas
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ematth7
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  • You are doing right. One can also give a nice geometrical proof of that formula, see here: https://en.wikipedia.org/wiki/Triangular_number#Relations_to_other_figurate_numbers. – Intelligenti pauca Sep 14 '15 at 21:01
  • After you add, you should be able to simplify the answer. A related question is http://math.stackexchange.com/questions/1128446/why-do-consecutive-triangular-numbers-in-pairs-like-6-and-10-always-add-up-t – David K Sep 14 '15 at 21:22

1 Answers1

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I'll assume that by "Tn-1" you mean "$T_{n-1}$" and not "$T_n-1$". (You could have produced this using $T_{n-1}$.)

I suspect that the idea isn't to use that formula and manipulate it algebraically, but to write the two sums like this:

  1  +  2  +  3  +...+(n-1)+ n
(n-1)+(n-2)+(n-3)+...+  1  + 0

and to conclude that they add up to $n^2$.

joriki
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