Use the Principle of Mathematical Induction to show that the

Use the Principle of Mathematical Induction to show that the following statement is true for all natural numbers n: 8+16+24+...+8n=4n(n+1)

Solution

statement 8 +16+24+....+8n=4n(n+1)

for n =1

8 =4*1*(1+1)

8=8 true

assume that statement if true for n =k

8 +16+24+....+8k=4k(k+1)

now for n =k+1

8 +16+24+....+8k+8(k+1)

=4k(k+1) +8(k+1)

=(k+1)(4k+8)

=(k+1)(4(k+2))

=4(k+1)((k+1)+1)

statement is true for n =k+1

therefore by principle of mathematical induction 8 +16+24+....+8n=4n(n+1)