Use induction to figure out the number of handshakes in a party Every arriving guest shakes hand with everybody else at a party. If there are n guests in the party, how many handshakes were there? Proof by using induction.
My approach to this problem was to write down a list of values for n and the corresponding people shaking hands. For instance, for n = 4, let's say A shakes hand with B, C, and D; B shakes hand with C and D; and C shakes hand with D.
Is the approach correct?

+---------+----------------------------------------------------+
|    n    | Number of handshakes                               |
+---------+----------------------------------------------------+
|    1    | Zero                                               |
|    2    | One                                                |
|    3    | A shakes hand with B and C; B shakes hand with C   |
|         | Therefore, 3 = 2 + 1                               |
|    4    | A shakes hand with B, C, and D;                    |
|         | B shakes hand with C and D;                        |
|         | C shakes hand with D;                              |
|         | Therefore, 6 = 3 + 2 + 1                           |
|    k    | (k-1) + (k-2) + ... + 2 + 1                        |
|   k+1   | k + (k-1) + ... + 2 + 1                            |
+---------+----------------------------------------------------+

What should I do next?


Try to derive the statement for $k+1$ using the truth of the statement for $k$:
Have a new guest, the $(k+1)-st$ guest arrive and shake everyone else's hand, after all the other $k$ guests have shaken hands with each other.  This last guest will shake hands with the remaining $k$ people, so that the total number of handshakes is the number of shakes between the previous $k$ guests, which you know, by the induction hypothesis is ... plus the shakes between the new guest and all the others guests.
For a different approach, notice that every pair of people shake hands, so all you need to do is to count the number of pairs present, i.e., the number of 2-element subsets from all the members.
