wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

If r is a fixed positive integer, prove by induction that (r+1)(r+2)(r+3)....(r+n) is divisible by n!


A

I want to see the solution

No worries! We‘ve got your back. Try BYJU‘S free classes today!
B

Take me to next question

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
Open in App
Solution

The correct option is B

Take me to next question


Let P(n): (r+1)(r+2)(r+3)....(r+n)=n!.k,

Where k is an integer.

When n=1,r+1=1!.(r+1)

P(1) is true.

let P(m) be true,i.e.,

(r+1)(r+2)(r+3)....(r+m)=m!.k ....(1)

Now, (r+1)(r+2)(r+3)....(r+m)(r+m+1)

= r(r+1)(r+2)(r+3)....(r+m)+(m+1)(r+1)(r+2)(r+3)...(r+m)

=(r+m)!(r1)!+(m+1)(m!)k,using(1)

=(m+1)!.(r+m)!(r1)!(m+1)!+(m+1)!.k

=(m+1)!.(r+mCr1+k)

=(m+1)!(integer+k)

P(m+1) is true;
Thus, p (m) is true
P (m+1) is true:

Thus,P(m) is true P(m+1)is true.

P(n) is true for all nϵN.


flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Mathematical Induction
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon