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Add by using the Horizontal method ${m^v} + n - 1{\text{ and }}{n^v} + 2{m^v} + n$.

Last updated date: 13th Jun 2024
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Hint: Horizontal method is the method for adding two or more algebraic equations or expressions. In this, equations or expressions are written in horizontal direction and similar terms are grouped and then added subsequently.

Complete step-by-step answer:
Firstly, write the expressions given in the question,
\[{m^v} + n - 1{\text{ and }}{n^v} + 2{m^v} + n\]
Then, horizontally, group the expressions in parentheses with + sign in between, as below
\[ = ({m^v} + n - 1) + ({n^v} + 2{m^v} + n)\]
After doing this, arrange the similar terms with signs in one space and group them, like as
\[ = ({m^v} + 2{m^v}) + {n^v} + (n + n) - 1\]
Now, add the similar terms in parentheses with signs. In this way, you will get the answer to the question asked above,
\[ = 3{m^v} + {n^v} + 2n - 1\]
Hence, the above expression is the solution.

Note: Identify the signs and powers before grouping the similar expressions. Though, the method explained above is Horizontal method for addition of algebraic equations or expressions but there is another method for addition of such problems is Column Method.
In this Column Method, each expression is written in a separate row below to each other and expressions are arranged in such a way that terms of each expression should come below to others in a column. After arranging this way, addition of the term is done column wise.
The question asked, now, can be solved by Column Method in the following manner.
Firstly, write the expressions given in the question,
\[{m^v} + n - 1{\text{ and }}{n^v} + 2{m^v} + n\]
Then, arrange these in a separate row one below to other and arrange the similar term with signs below to other as,
$ {m^v} + n - 1 \\
  2{m^v} + n + {n^v} \\ $
Now, add the similar terms. If you find the terms which are not similar to each other then write those terms as same as they are with their signs,
  {m^v} + n - 1 \\
  2{m^v} + n + {n^v} \\
 }{{3{m^v} + 2n + {n^v} - 1}}$
Thus, the solution to the question is $3{m^v} + 2n + {n^v} - 1$