Question

Is there an \[{n^{th}}\] term of the sequence \[1,5,10,15,20,25?\]

Answer 1

Hint: Here, we use some concepts of Arithmetic Progression to solve this problem. We will also learn about a progression and all the terminology present in that. And we use another formula, which is \[{a_n} = a + (n - 1)d\] , which gives us the \[{n^{th}}\] term of Arithmetic Progression and then generalize a progression.

Complete step by step solution:
An Arithmetic Progression (AP) is a set of numbers in which the difference between consequent terms is a constant term.
For example, consider a progression \[3,7,11,15,19,......\]
So, here, the difference between a term and its next term is 4 which is a constant. So, this is an AP.
And the constant is known as “common difference” generally represented by \[d\] .
And the first term is denoted by \[a\] .
And \[{n^{th}}\] term is equal to \[a + (n - 1)d\] .
So, finally, generalised AP is written as follows
\[a,(a + d),(a + 2d),.......(a + (n - 1)d).....\]
So, now, in the question, the sequence given is \[1,5,10,15,20,25\]
The difference between first and second terms is \[5 - 1 = 4\]
The difference between the second and the third terms is \[10 - 5 = 5\]
The difference between third and fourth terms is \[15 - 10 = 5\]
As we observe here, there is no constant difference.
So, the given sequence is NOT an AP.
Therefore, there is no \[{n^{th}}\] term for this sequence.

Note:
Here, the whole sequence is not an Arithmetic progression, but when we remove the first term, then the sequence will become an Arithmetic progression because there is a constant difference between consequent terms. That means \[5,10,15,20,25\] is an AP.
Because \[10 - 5 = 15 - 10 = 20 - 15 = 25 - 20 = 5{\text{ (common difference)}}\]
All the numbers in an AP are called as terms. In an AP, the value of the first term can also be a negative value. And similarly, the common difference can also be a negative value. A term of an AP is equal to the average of the terms succeeding and preceding it. For example, if \[{a_1},{a_2},{a_3}\] are in AP, then \[{a_2} = \dfrac{{{a_1} + {a_3}}}{2}\] . We use this formula to solve many problems.