Chapter A.2, Problem 11E

(a)

To determine

Prove that the finite sum rule ${\displaystyle \sum _{k=1}^n\left(a_k+b_k\right)}={\displaystyle \sum _{k=1}^n{a}_k}+{\displaystyle \sum _{k=1}^n{b}_k}$ holds for every positive integer n.

(b)

To determine

Prove that the finite sum rule ${\displaystyle \sum _{k=1}^n\left(a_k-b_k\right)}={\displaystyle \sum _{k=1}^n{a}_k}-{\displaystyle \sum _{k=1}^n{b}_k}$ holds for every positive integer n.

(c)

To determine

Prove that the finite sum rule ${\displaystyle \sum _{k=1}^{n}c{a}_k}=c\cdot {\displaystyle \sum _{k=1}^n{a}_k}$ holds for every positive integer n.

(d)

To determine

Prove that the finite sum rule ${\displaystyle \sum _{k=1}^n{a}_k}=n\cdot c$ holds for every positive integer n if $a_k$ has the constant value c.