Mathematical Induction

Question

Use mathematical induction to prove that each statement is true for every positive integer n:

 

 

2 + 7 + 12 + . . . + (5n-3) = n(5n-1)/2

Answer 1

it is true for x = 1

5(1) - 3 = (1)[5(1) - 1]/2

5-3 = 4/2

2 = 2

 

If true for n, is it true for n+1?

n(5n-1)/2 = n(5n-1)/2

 

n(5n-1)/2 + [5(n+1) -3] = (n + 1)[5 (n+1) -1]/2

 

(5n² - n)/2 + (5n + 5 -3) = [(5n² + 10n +5) - (n+1)]/2

 

(5/2)n² + (9/2)n + 2 = (5n² + 9n +4)/2

 

for ease of calculation and recognition, just multiply both sides by 2

 

5n² + 9n + 4 = 5n² + 9n + 4

 

∴ if true for n, then true for n+1