Question 1115660: use mathematical induction to prove that each statement is true for all positive integers.
0+7+14+...+7n-7=(n(7n-7)/2
Answer by math_helper(2461)   (Show Source):
You can put this solution on YOUR website! n=1: LHS=0, RHS= 0*(7*0-7)/2 = 0 (ok for n=1)
n=2: 0+7 = 7 , 2(7*2-7)/2 = 2*7/2 = 7 (ok for n=2)
Assume this is true for n=k:
0+7+14+…+(7k-7) = k(7k-7)/2
Let n=k+1, need to show 0+7+14+…+(7(k+1)-7) = (k+1)(7(k+1)-7)/2
LHS = [ 0+7+14+…+(7(k+1)-7) = 0+7+14+…+(7k-7) ] + (7(k+1)-7)
Using induction hypothesis for the bracketed part:
LHS = [ k(7k-7)/2 ] + (7(k+1)-7)
= k(7k-7)/2 + 7k
= k(7k-7)/2 + 14k/2
= (7k^2 - 7k + 14k)/2
= (7k^2 + 7k)/2
= [7(k)](k+1) / 2
= [(7(k+1)-7)](k+1) / 2
Done.
Since the relationship holds for n=k+1, it holds for all positive integers.
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