Question 1081848
{{{ S[1] = 3  }}}
{{{ S[2] = 9 = 3*(2)*(2+1)/2 }}}
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Assume {{{ S[k] = 3*k(k+1)/2 }}}   (*)

What is {{{ S[k+1] }}} ?

It is the previous sum plus 3*(k+1):
{{{ S[k+1] = S[k] + 3(k+1) }}}
{{{ S[k+1] = 3k(k+1)/2 + 3(k+1) }}}
{{{ S[k+1] = 3(k+1)*(k/2 + 1)}}}
{{{ S[k+1] = 3(k+1)*(k+2)/2 }}}   <<<— <<<  DONE  (same form as (*) )

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If the above is not clear,  let  h=k+1,  then {{{ S[k+1] = S[h] = 3h(h+1)/2 }}} which should make it clear.