Question

Find n if n - 2, 4n - 1 and 5n + 2 are in AP.

Answer 1

Hint: As we are given that the terms are in AP then it is clear that there exist a common difference between the terms .hence from this condition we get that ${t_2} - {t_1} = {t_3} - {t_2}$ and now substituting the given terms we can obtain the value of n.

Complete step by step solution:
We are given that $n - 2,4n - 1$ and $5n + 2$are in AP
We know that the terms in an arithmetic sequence have a common difference d
The common difference is obtained by subtracting the first term from the second term
$ \Rightarrow {t_2} - {t_1}$
Since it is in AP we know that it has a common difference
Hence
$ \Rightarrow {t_2} - {t_1} = {t_3} - {t_2}$…………(1)
Here our first term is $n - 2$
The second term is $4n - 1$
And the third term is $5n + 2$
Using this in (1) we get
$
   \Rightarrow 4n - 1 - \left( {n - 2} \right) = 5n + 2 - \left( {4n - 1} \right) \\
   \Rightarrow 4n - 1 - n + 2 = 5n + 2 - 4n + 1 \\
   \Rightarrow 3n + 1 = n + 3 \\
   \Rightarrow 3n - n = 3 - 1 \\
   \Rightarrow 2n = 2 \\
   \Rightarrow n = \dfrac{2}{2} = 1 \\
 $
Hence we get the value of $n$ is 1.

Note:
1) An arithmetic sequence is a list of numbers with a definite pattern. If you take any number in the sequence then subtract it by the previous one, and the result is always the same or constant then it is an arithmetic sequence.
2) The constant difference in all pairs of consecutive or successive numbers in a sequence is called the common difference, denoted by the letter d. We use the common difference to go from one term to another.Take the current term and add the common difference to get to the next term, and so on. That is how the terms in the sequence are generated.