Prove that these two series are equal

[chwala]

Homework Statement

The question is attached as it appears on the textbook.

Relevant Equations

sum of series

i looked at this and it was not making any sense at all, could it be a textbook error or i am missing something here; note that,
lhs gives us, $4,6,8,10,12,14$
rhs gives us, $8,11,14,17,20,23$

 

[anuttarasammyak]

$$LHS=\sum_{k=1}^6 (2(k+2)-2)=\sum_{k=1}^6 (2k+2)$$
$$RHS=\sum_{k=1}^6 (3k+5)$$
So
$$RHS-LHS=\sum_{k=1}^6 (k+3)= \frac{6*7}{2}+3*6=39\neq 0$$

 

[chwala]

$$LHS=\sum_{k=1}^6 (2(k+2)-2)=\sum_{k=1}^6 (2k+2)$$
$$RHS=\sum_{k=1}^6 (3k+5)$$
So
$$RHS-LHS=\sum_{k=1}^6 (k+3)= \frac{6*7}{2}+3*6=39\neq 0$$

therefore there is an error on the textbook...

 

[chwala]

Ok, then what could have been the right question?...,assuming that an error was made. How do we modify the original question in order for the proof to hold (i am assuming that the error made was on the right hand side of the equation)...

 

[etotheipi]

You can try and work that out yourself! Use the substitution $u = k-2$ to re-write the sum on the LHS

 

[chwala]

You can try and work that out yourself! Use the substitution $u = k-2$ to re-write the sum on the LHS

has that not been done in post number $2$?