Suppose we want to prove the statement S(n): \"If n 2, the sum of the integers 2

Question

Suppose we want to prove the statement S(n): "If n 2, the sum of the integers 2 through n is (n+2)(n-1)/2" by induction on n. To prove the inductive step, we can make use of the fact that 2+3+4+...+(n+1) = (2+3+4+...+n) + (n+1) Find, in the list below an equality that we may prove to conclude the inductive part. a) If n 3 then (n+2)(n-1)/2 + n + 1 = (n+3)(n)/2 b) If n 1 then (n+2)(n-1)/2 + n + 1 = (n+3)(n)/2 c) If n 2 then (n+2)(n-1)/2 + n + 1 = (n+3)(n)/2 d) If n 1 then (n+2)(n-1)/2 + n + 1 = n(n+3)/2 Suppose we want to prove the statement S(n): "If n 2, the sum of the integers 2 through n is (n+2)(n-1)/2" by induction on n. To prove the inductive step, we can make use of the fact that 2+3+4+...+(n+1) = (2+3+4+...+n) + (n+1) Find, in the list below an equality that we may prove to conclude the inductive part. a) If n 3 then (n+2)(n-1)/2 + n + 1 = (n+3)(n)/2 b) If n 1 then (n+2)(n-1)/2 + n + 1 = (n+3)(n)/2 c) If n 2 then (n+2)(n-1)/2 + n + 1 = (n+3)(n)/2 d) If n 1 then (n+2)(n-1)/2 + n + 1 = n(n+3)/2 Suppose we want to prove the statement S(n): "If n 2, the sum of the integers 2 through n is (n+2)(n-1)/2" by induction on n. To prove the inductive step, we can make use of the fact that 2+3+4+...+(n+1) = (2+3+4+...+n) + (n+1) Find, in the list below an equality that we may prove to conclude the inductive part. a) If n 3 then (n+2)(n-1)/2 + n + 1 = (n+3)(n)/2 b) If n 1 then (n+2)(n-1)/2 + n + 1 = (n+3)(n)/2 c) If n 2 then (n+2)(n-1)/2 + n + 1 = (n+3)(n)/2 d) If n 1 then (n+2)(n-1)/2 + n + 1 = n(n+3)/2

Explanation / Answer

Hi,

First to answer this, lets understand the concept of mathematical induction, there are 3 steps invlolved.
1. prove for n=1

2. assume for n its true

3. prove it holds for n+1
now, for the 3rd step we have LHS as 2+3+..+ (n+1) = (2+3+..+n) + n+1
now,since we assumed its true for n in 2nd step we can replace the bracket in RHS by (n+2)(n-1)/2,
if we do that we get
(n+2)(n-1)/2+(n+1)
= (n^2-n+2n-2+2n+2)/2
=(n^2+3n)/2
=n(n+3)/2
which is 3rd option.
Thumbs up if this was helpful, otherwise let me know in comments.

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.