Question

Identify the pattern with which each number in the list of numbers \[12,23,34,45,56,...\;\] is formed.
A. New number \[ = \] preceding number \[ + 11\]
B. New number \[ = \] preceding number \[ + 12n - 1\]
C. New number \[ = \] preceding number \[11n\]
D. There is no specific pattern between its numbers.

Answer 1

Hint:In the above given question, we are given a list of numbers written as \[12,23,34,45,56,...\;\] . Clearly given list of numbers is a sequence of natural numbers. As we can see the first term of this sequence is \[12\] , the second term is then \[23\] , and so on. We have to find the pattern involved in the given sequence. In order to approach the solution, we can start by looking for the difference between two consecutive terms of the given sequence to find the pattern that is involved in the above given sequence.

Complete step by step answer:
Given sequence is \[12,23,34,45,56,...\;\] We need to find the pattern of the above sequence. First we should check the difference between two consecutive terms of this sequence and see if they have a similar pattern in each difference or not.Now, as it is given that the first term of the sequence is \[12\] and the second term is \[23\].
Hence we can see that the difference of these two terms is \[23 - 12 = 11\] .
Also, the difference between the second and the third term is \[34 - 23 = 11\] .

Similarly, the difference between the third and the fourth term is \[45 - 34 = 11\] .
And the difference between the fourth and the fifth term is \[56 - 45 = 11\] .
Therefore, we have found that the difference of any two consecutive terms of this sequence is equal to \[11\] . Thus, that means each next term of this sequence is \[11\] more than its preceding term. Hence the pattern involved in this sequence is given by,
New number \[ = \] preceding number \[ + 11\]

So the correct option is A.

Note:Since in the given sequence \[12,23,34,45,56,...\;\] , each next term is \[11\] more than its preceding term i.e. the difference between any two consecutive terms of this sequence is constant and is equal to \[11\] . Now, since the first term of this sequence is \[12\] , therefore we can say that the given list of numbers is nothing else but an Arithmetic Progression i.e. AP where the first term is \[12\] and the common difference is \[11\] .
Hence, given AP can be written as,
\[ \Rightarrow 12,23,34,45,56,67,78,89,100...\;\]
\[ \Rightarrow {a_n} = a\left( {n + 1} \right)d\]
Where \[a = 12\] , \[d = 11\] and \[{a_n}\] is the \[nth\] term of the AP.