Using inductive reasoning, write a formula for the number of handshakes if the number of people is n and each?

Question

person shakes hands with one another...

if seven people shake hands it is 21 right 6+5+4+3+2+1 ???

Answer 1

so for n people, number of handshakes is 1+2+3...+ (n-1)

so this is arithmetic sequence so sum of it is number of handshakes

S= ((n-1)/2 * (1+ (n-1))

S= (n^2-n) / 2  

now prove that 1+2+3+ (n-1) = (n^2-n) / 2 using mathematical induction

prove it is true using n= 1

0 = 0^2-0/ 2

0=0

so assume it is true for all integer k

S= (k^2-k) /2

now prove it is true for (k+1)

S(k+1) = (k^2-k)/2 + (k+1-1)

= (k^2 - k)/2 + k

= (k^2 - k)/2 + 2k/2

= (k^2 +k)/ 2

= (k^2+ k + k -k + 1 -1)/2

= (k^2+ 2k +1 - (k+1))/2

= ((k+1)^2 + (k+1)) /2

and since it is also true for k+1 then it must be true for all positive integers n

hope this helps