Question

http://imageshack.us/photo/my-images/824/page679.jpg/
I need help number 21
thanks

Answer 1

One can use induction for it.Check if it is valid for the first term, p(1).
LHS:
p(1)=1
RHS:
p(1)=(7^1 -1)/6=(7-1)/6=1
LHS=RHS.
Therefore you can use mathematical induction to prove the statement.
Assume p(n) is true.
Now using that prove that p(n+1)=(7^(n+1)-1)/6
LHS:
1+7+49..+7^(n-1)+7^n
=(7^n-1)/6 +7^n
=((7^n-1)+6*7^n)/6
=(7^n(1+6)-1)/6
=(7^(n+1)-1)/6
RHS:
p(n+1)=(7^(n+1)-1)/6
LHS=RHS