# PROBLEMS ON MATHEMATICAL INDUCTION

## About "Problems on Mathematical Induction"

Problems on Mathematical Induction :

Here we are going to see some mathematical induction problems with solutions.

Define mathematical induction :

Mathematical Induction is a method or technique of proving mathematical results or theorems

The process of induction involves the following steps.

## Problems on Mathematical Induction

Question 1 :

Using the Mathematical induction, show that for any natural number n ≥ 2,

[1/(1 + 2)] + [1/(1 + 2 + 3)] + [1/(1 + 2 + 3 + 4)]  + · · · + [1/(1 + 2 + 3 + · · · + n)]  =  (n − 1)/(n + 1)

Solution :

Let p(n) =  [1/(1 + 2)] + [1/(1 + 2 + 3)] + [1/(1 + 2 + 3 + 4)]  + · · · + [1/(1 + 2 + 3 + · · · + n)]   =  (n − 1)/(n + 1)

Step 1 :

put n = 2

p(2)  =  p(n) =  [1/(1 + 2)]  =  (2 − 1)/(2 + 1)

1/3  =  1/3

Hence p(2) is true.

Step 2 :

Let us assume that the statement is true for n = k

p(k) =  [1/(1 + 2)] + [1/(1 + 2 + 3)] + [1/(1 + 2 + 3 + 4)]  + · · · + [1/(1 + 2 + 3 + · · · + k)]   =  (k − 1)/(k + 1)  ----(1)

We need to show that P(k + 1) is true. Consider,

Step 3 :

Let us assume that the statement is true for n = k + 1

p(k+1)

p(n) =  [1/(1 + 2)] + [1/(1 + 2 + 3)] + [1/(1 + 2 + 3 + 4)]  + · · · + [1/(1 + 2 + 3 + · · · + (k+1))]   =  (n − 1)/(n + 1)

By applying (1) in this step, we get

k(k+1)/(k+1)(k+2)  =  k/(k+2)

k/(k+2)  =  k/(k+2)

Hence, by the principle of mathematical induction  n ≥ 2,

[1/(1 + 2)] + [1/(1 + 2 + 3)] + [1/(1 + 2 + 3 + 4)]  + · · · + [1/(1 + 2 + 3 + · · · + n)]  =  (n − 1)/(n + 1)

Question 2 :

Using the Mathematical induction, show that for any natural number n,

[1/(1.2.3)]+[1/(2.3.4)]+[1/(3.4.5)]+ · · · +[1/(n.(n + 1).(n + 2))]

=  n(n + 3)/4(n + 1)(n + 2)

Solution :

Let p(n)  = [1/(1.2.3)]+[1/(2.3.4)]+[1/(3.4.5)]+ · · · +[1/(n.(n + 1).(n + 2))]  =  n(n + 3)/4(n + 1)(n + 2)

Step 1 :

Put n = 1

[1/(1.(1 + 1).(1 + 2))]  =  1(1 + 3)/4(1 + 1)(1 + 2)

1/(1.2.3)  =  1(4)/4(2)(3)

1/(1.2.3) =  1/(1.2.3)

Hence p(1) is true.

Step 2 :

Let us assume that the statement is true for n = m

[1/(1.2.3)]+[1/(2.3.4)]+[1/(3.4.5)]+ · · · +[1/(m.(m+1).(m+2))]  =  m(m + 3)/4(m + 1)(m + 2)  --(1)

We need to show that P(m + 1) is true. Consider,

Step 3 :

Let us assume that the statement is true for n = m + 1

p(m+1)

By expanding L.H.S, we get

=  (m(m2+6m+9) + 4)/4(m+1)(m+2)(m+3)

=  (m3 + 6m+ 9m + 4)/4(m+1)(m+2)(m+3)

=  (m+1)2(m+4)/4(m+1)(m+2)(m+3)

=  (m+1)(m+4)/4(m+1)(m+2)(m+3)  --->R.H.S

Hence, by the principle of mathematical induction [1/(1.2.3)]+[1/(2.3.4)]+[1/(3.4.5)]+ · · · +[1/(n.(n + 1).(n + 2))]

=  n(n + 3)/4(n + 1)(n + 2)

After having gone through the stuff given above, we hope that the students would have understood "Problems on Mathematical Induction"

Apart from the stuff given above, if you want to know more about "Problems on Mathematical Inductions". Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

## Recent Articles

1. ### SAT Math Videos

May 22, 24 06:32 AM

SAT Math Videos (Part 1 - No Calculator)

2. ### Simplifying Algebraic Expressions with Fractional Coefficients

May 17, 24 08:12 AM

Simplifying Algebraic Expressions with Fractional Coefficients