Question 1 Exercise 7.1

Solutions of Question 1 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, $2+4+6+\cdots+2 n=n(n+1)$

1. For $n=1$ then $$2=1(1+1)=2 $$ Hence the above proposition is true for $n=1$. 2. Let it be true for $n=k$, then $$2+4+6+\cdots+2 k=k(k+1)....(i)$$ 3. For $n=k+1$, the $(k+1)^{t h}$ term of the series, which is: $$a_{k+1}=\mathbf{2}(k+1)=2 k+2 $$ Adding this $k+1$ term to both sides of the induction hypothesis (i) \begin{align}2+4+6+\cdots+2 k+2(k+1)& =k(k+1)+2(k+1) \\ & =(k+1)[k+2] \\ & =(k+1)(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 $n \in \mathbf{N}$.