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

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

Explanation

Solution

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 an=a+(n−1)d{a_n} = a + (n - 1)d , which gives us the nth{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,......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 dd .
And the first term is denoted by aa .
And nth{n^{th}} term is equal to a+(n−1)da + (n - 1)d .
So, finally, generalised AP is written as follows
a,(a+d),(a+2d),.......(a+(n−1)d).....a,(a + d),(a + 2d),.......(a + (n - 1)d).....
So, now, in the question, the sequence given is 1,5,10,15,20,251,5,10,15,20,25
The difference between first and second terms is 5−1=45 - 1 = 4
The difference between the second and the third terms is 10−5=510 - 5 = 5
The difference between third and fourth terms is 15−10=515 - 10 = 5
As we observe here, there is no constant difference.
So, the given sequence is NOT an AP.
Therefore, there is no nth{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 5,10,15,20,255,10,15,20,25 is an AP.
Because 10−5=15−10=20−15=25−20=5 (common difference)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 a1,a2,a3{a_1},{a_2},{a_3} are in AP, then a2=a1+a32{a_2} = \dfrac{{{a_1} + {a_3}}}{2} . We use this formula to solve many problems.