Question 2 Exercise 7.1

Solutions of Question 2 of Exercise 7.1 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

Establish the formulas below by mathematical induction, $1+5+9+\ldots+(4 n-3)=n(2 n-1)$

1. For $n=1$ then $$1=1(2.1-1)=1$$ Hence the above proposition is true for $n=1$.

2. Let it be true for $n=k$, then \begin{align}1+5+9+\ldots+(4 k-3)\\ & =k(2 k-1)....(i) \\ \end{align} 3. For $n=k+1$, the $k+1$ term of the series, which is: $$a_{k-1}=4(k+1)-3=4 k+1 $$ Adding this $(k+1)^{t h}$ term to both sides of the induction hypcthesis (i) \begin{align}1+5+9+\ldots+(4 k-3)+(4 k+1)& =k(2 k-1)+4 k+1 \\ & =2 k^2-k+4 k+1 \\ & =2 k^2+3 k+1 \\ & =2 k^2+2 k+k+1 \\ & =2 k(k+1)+1(k+1) \\ \Rightarrow 1+5+9+\ldots+(4 k-3)+(4 k+1)& =(k+1)[2 k+1] \\ & =(k+1)[2(k+1)-1]\end{align} Which is the form taken by proposition when $n$ is replaced by $k+1$. hence it is true for $n=k+1$.

Thus by mathematical induction it it true for $n \in \mathbf{N}$.