Question
Use mathematical induction to prove that the following is true.
8+11+14...+(3n+5)=1/2n(3n+13), for all n in the set of natural numbers.
8+11+14...+(3n+5)=1/2n(3n+13), for all n in the set of natural numbers.
Answers
Steve
prove that P(1) is true:
8 = 1/2 1(3*1+13) = 16/2 = 8
Assuming P(k), see what P(k+1) means:
8+11+...+(3k+5)+(3(k+1)+5) = k/2 (3k+13) + (3(k+1)+5)
= k/2 (3k+13) + 3k+8
1/2 (3k^2+13k + 6k+16)
= 1/2 (3k^2+19k+16)
= 1/2 (k+1)(3k+16)
= 1/2 (k+1)(3(k+1)+13)
= P(k+1)
So, P(1) and P(k) ==> P(k+1)
8 = 1/2 1(3*1+13) = 16/2 = 8
Assuming P(k), see what P(k+1) means:
8+11+...+(3k+5)+(3(k+1)+5) = k/2 (3k+13) + (3(k+1)+5)
= k/2 (3k+13) + 3k+8
1/2 (3k^2+13k + 6k+16)
= 1/2 (3k^2+19k+16)
= 1/2 (k+1)(3k+16)
= 1/2 (k+1)(3(k+1)+13)
= P(k+1)
So, P(1) and P(k) ==> P(k+1)