Question

If a number sequence begins 1, 3, 4, 6, 7, 9, 10, 12…………………..., which of the following numbers does not appear in the sequence?
a) 34
b) 43
c) 57
d) 65
e) 72

Answer 1

Hint: Try to figure out that the sequence mentioned in the question is a combination of two different arithmetic progressions, and if the number given in the options is part of even one of the two A.P., then it is a part of the sequence too.

Complete step-by-step answer:

Before starting with the solution, let us discuss what an A.P. is. A.P. stands for arithmetic progression and is defined as a sequence of numbers for which the difference of two consecutive terms is constant. The general term of an arithmetic progression is denoted by ${{T}_{r}}$, and sum till n terms is denoted by ${{S}_{r}}$ .

${{T}_{r}}=a+\left( r-1 \right)d$

${{S}_{r}}=\dfrac{r}{2}\left( 2a+\left( r-1 \right)d \right)$

Now moving to the sequence given in the question, we can say that it is a combination of two arithmetic progressions: 1, 4, 7, 10,…….., and 3 ,6 ,9 , 12……….. From here, we can also conclude that all the positive multiples of 3 are the part of the sequence and all the numbers of the type 3k+1, where k is positive whole numbers are also the part of the sequence.

Now we will check the options to find which number among the options is not the part of the series. So, starting with option (a), i.e., 34, and we know that 34 is a number of 3k+1 type with k=11. Hence, it is part of the sequence.

Now moving to option (b), and we know that 43 is also a number of 3k+1 type, where k=14. So, it is also part of the sequence. If we see option (c), 57 is a multiple of 3, making it a part of the given sequence.

Now when we check option (d), we find that 65 is neither a multiple of 3, nor a 3k+1 type number.

Therefore, we can conclude that the answer to the above question is option (d).

Note: It is not always necessary that you can break the sequences in the form of simple arithmetic progressions and draw results, but it is for sure that the terms of a sequence will have some order, and it depends on you how wisely you figure out the pattern. The suggested approach can be to break in two or more simple sequences or to try to find the general term of the sequence by observing the pattern of numbers appearing in the sequence.