Answer

Verified

447k+ views

**Hint:**Here in this question the concept of mathematical induction will get used which states that if the statement is true for n=k then it will also be true for its successor i.e. n=k+1. This is known as principle of mathematical induction

**Complete step-by-step answer:**

Let the given statement be named as P(n) so it can be written as$P(n) = 1 + 4 + 7 + .... + (3n - 2) = \dfrac{1}{2}n(3n - 1)$, $n \in N$

For n=1

$ \Rightarrow P(1) = \dfrac{1}{2}(1)(3 - 1) = 1$ which is true

Now we will assume that P(m) is true for some positive integers ‘m’$ \Rightarrow P(m) = 1 + 4 + 7 + .... + (3m - 2) = \dfrac{1}{2}m(3m - 1)$

Now according to principle of mathematical induction if a statement is true for n=m then it will also be true for its successor i.e. n=m+1, therefore we shall prove that P(m+1) is also true.$ \Rightarrow P(m + 1) = 1 + 4 + 7 + .... + (3(m + 1) - 2) = \dfrac{1}{2}(m + 1)((3m + 3) - 1)$$ \Rightarrow 1 + 4 + 7 + .... + (3m + 3 - 2) = \dfrac{1}{2}(m + 1)(3m + 2)$ $ \Rightarrow 1 + 4 + 7 + .... + (3m + 1) = \dfrac{1}{2}(m + 1)(3m + 2)$

L.H.S=$1 + 4 + 7 + .... + (3m + 1)$

R.H.S=$\dfrac{1}{2}(m + 1)(3m + 2)$

Now we will prove L.H.S=R.H.S

$ \Rightarrow 1 + 4 + 7 + .... + (3m - 2) + (3m + 1)$ (Adding term before 3m+1)

Here we will apply formula of sum of arithmetic progression i.e. ${S_n} = \dfrac{n}{2}(2a + (n - 1)d)$ $ \Rightarrow {S_n} = \dfrac{m}{2}(2(1) + (m - 1)(4 - 1))$ (Here a=1, d=4-1 and m is the total number of terms)

Now putting and solving the equation we will get a sum of arithmetic progression. $ \Rightarrow {S_n} = \dfrac{m}{2}(2 + (m - 1)3)$

$ \Rightarrow {S_n} = \dfrac{m}{2}(2 + 3m - 3) = \dfrac{m}{2}(3m - 1)$

$ \Rightarrow \dfrac{{m(3m - 1)}}{2} + (3m + 1)$

Now taking L.C.M we will get

$ \Rightarrow \dfrac{{m(3m - 1) + 2(3m + 1)}}{2}$

$ \Rightarrow \dfrac{{3{m^2} - m + 6m + 2}}{2}$

$ \Rightarrow \dfrac{{3{m^2} + 5m + 2}}{2} = \dfrac{{3{m^2} + 3m + 2m + 2}}{2}$

(Splitting the equation to find the factors of quadratic equation)

$ \Rightarrow \dfrac{{3m(m + 1) + 2(m + 1)}}{2}$

$ \Rightarrow \dfrac{{(3m + 2)(m + 1)}}{2}$=R.H.S

Therefore P(m+1) holds true.

Thus, by principle of mathematical induction, for all $n \in N$, P(n) holds true.

**Note:**Some students may find difficulty in splitting the quadratic equation whose coefficient of degree two variable is not one so below is the explanation of it: -

Let’s take above equation as an example: -$3{m^2} + 5m + 2$

Now comparing with general quadratic equation $a{x^2} + bx + c$

Here a=3, b=5, c=2

Step1. Multiply ab i.e. $ab = 3 \times 2 = 6$

Step2. Now split 6 into that factors so that its sum is equal to ‘b’

$ \Rightarrow 3 \times 2 = 6$ Factors are 3 and 2 whose sum is 5.

Therefore 5m will get split into 3m and 2m and the equation will become $3{m^2} + 3m + 2m + 2$

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

The polyarch xylem is found in case of a Monocot leaf class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Change the following sentences into negative and interrogative class 10 english CBSE

Casparian strips are present in of the root A Epiblema class 12 biology CBSE