Important Questions for Class 11 Chapter 4 - Principal of Mathematical Induction

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Important questions based on NCERT syllabus Class 11 Chapter 4-Principal of Mathematical Induction:

Question 1: Prove by using mathematical induction that sum of cubes of the first n natural numbers is
1^3 + 2^3 +....+n^3 = (n(n+1)/2)^2

Solution:

Let the given statemnet be p(n) i.e., (n(n+1)/2)^2
for n = 1
(1(1+1)^2/2)^2 = (1.2/2)^2 = 1^2 = 1
p(1) = 1^3 = 1, which is true

Let p(k) be true for some positive integer k, i.e.,
1^3 + 2^3 +....+k^3 = (k(k+1)/2)^2 ..........(1)
Now we prove that p(k+1) is also true

1^3 + 2^3 +....+k^3 +(k+1)^3 = k(k+1)/2)^2 + (k+1)^3
= k^2(k+1)^2/4 + (k+1)^3
= (k^2(k+1)^2 + 4(k+1)^3)/4
= ((k+1)^2{k^2+4(k+1)})/4
= ((k+1)^2{k^2+4k+4})/4
= ((k+1)^2(k+2)^2)/4
= ((k+1)^2(k+1+1)^2)/4
1^3 + 2^3 +....+k^3 +(k+1)^3 = (((k+1)(k+1+1))/2)^2

Thus p(k+1) is true whenever p(k) is true.
Hence by mathematical induction p(n) is true for all natural numbers.

Question 2: Prove that 1 + 1/(1+2) + 1/(1+2+3) + 1/(1+2+3+4) + ..... + 1/(1+2+3+...+ n) = 2n/(n+1) by using mathematical induction.

Solution:

Let the given statemnet be p(n) i.e., 2n/(n+1)
for n = 1
2n/(n+1) = 2(1)/(1+1)
= 2/2
= 1
which is true

Let p(k) be true for some positive integer k, i.e.,

1 + 1/(1+2) + 1/(1+2+3) + 1/(1+2+3+4) + ..... + 1/(1+2+3+...+ k) = 2k/(k+1)..........(1)

Now we prove that p(k+1) is also true

1 + 1/(1+2) + 1/(1+2+3) + 1/(1+2+3+4) + ..... + 1/(1+2+3+...+ k)+ 1/(1+2+3+...+ k + (k+1)

(1 + 1/(1+2) + 1/(1+2+3) + 1/(1+2+3+4) + ..... + 1/(1+2+3+...+ k))+ 1/(1+2+3+...+ k + (k+1)

2k/(k+1) + 1/(1+2+3+...+ k + (k+1) [using (1)]

2k/(k+1) + 1/((k+1)(k+1+1)/2) [ 1+2+3+...+n = n(n+1)/2)]

2k/(k+1) + 2/(k+1)(k+2)

2/k+1 (k+ 1/(k+2))