Question

When n=1 there is 1 dot. when n=2 there are 3 dots, when n=3 there are 6 dots. notice that the total number of dots increases by n each time.
use induction to prove that
d(n)= n(n+1)/2

part A: prove the statement is true for n=1


part B: Assume that the statement is true for n=k + 1, therefor proving it true for all natural numbers, n.

hint* since the total number of dots increase by n each time, prove that d(k) + (k+1) = d (k+1)


*please answer i really need help!!

Answer 1

Answer:

Part A

• d(n) = n(n + 1)/2
• n = 1 ⇒ d(1) = 1(1 + 1)/2 = 2/2 = 1, proved

Part B

• d(k) + (k + 1) =
• k(k + 1)/2 + (k + 1) =
• [k(k + 1) + 2(k + 1)]/2 =
• [(k + 1)(k + 1 + 1)]/2 =
• d(k + 1)

• Proved