A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely. Show your work. (refer to image)?

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1 Answer
May 1, 2018

#S_k = (k(k+1)(k+2))/3#
#S_(k+1) = ((k+1)(k+2)(k+3))/3#

Explanation:

We are given that #S_n = (n(n+1)(n+2))/3#.

To find #S_k#, simply replace all occurrences of #n# in #S_n# with #k#. That is, #S_k = (k(k+1)(k+2))/3#.

To find #S_(k+1)#, replace all instances of #n# in #S_n# with #k+1#. This gives #S_(k+1) = ((k+1)((k+1)+1)((k+1)+2))/3#. Combining terms, this gives #S_(k+1) = ((k+1)(k+2)(k+3))/3#.