Q. 58 - Calculus

Question

Given a simple proof that $\sum _{k=5}^n(a_k+b_k)=\sum _{k=5}^n{a}_k+\sum _{k=5}^n{b}_k$

Step-by-Step Solution

Answer

We prove the statement $\sum _{k=5}^n(a_k+b_k)=\sum _{k=5}^n{a}_k+\sum _{k=5}^n{b}_k$

Step 1: Given information

We are given a relation 

$\sum _{k=5}^n(a_k+b_k)=\sum _{k=5}^n{a}_k+\sum _{k=5}^n{b}_k$. We have to prove this

Step 2: Proof

Consider LHS

We have,

$$\begin{gathered} \sum _{k=5}^n(a_k+b_k) \\ =(a_5+b_5)+(a_6+b_6)+....+(a_n+b_n) \\ =(a_5+a_6+...+a_n)+(b_5+b_6+....+b_5) \\ =\sum _{k=5}^n{a}_k+\sum _{k=5}^n{b}_k \\ =RHS \end{gathered}$$

Hence proved