Question

Find the missing term in the following sequence.
$177,170,159,146,?,110$
A. $132$
B. $106$
C. $129$
D. $127$
E. None of these

Answer 1

Hint: To find the missing terms of the given sequence, we will find some pattern or some logic. In this example, we will make a positive difference of consecutive terms of the sequence to find the pattern or logic which is used.

Complete step by step answer:
Let us assume the missing term is $x$. Therefore, the given sequence is $177,170,159,146,x,110$.
Let us take the positive difference of consecutive terms of the given sequence. We get,
$
  177 - 170 = 7 \\
  170 - 159 = 11 \\
  159 - 146 = 13 \\
$
If we observe the above differences then we can see that $7,11,13$ are prime numbers. After $13$, the next prime numbers are $17$ and $19$. Therefore, we can say that the difference of $146$ and $x$ must be $17$. That is, $146 - x = 17$. That is, $x = 146 - 17 = 129$. Also we can see that the difference of $129$ and last term $110$ is $19$ which is the next prime number after $17$.
Therefore, in the given sequence the missing term is $129$.
Therefore, option C is correct.

Note:
There may be more than one way to find the missing term of number sequence. There are so many types of number sequences. In this example, we can say that the given sequence is a mixed sequence because a mixed sequence may have more than one pattern arranged in a single sequence. There is no fixed pattern always for mixed sequences. If the difference between two consecutive terms is equal (constant) then it is called an arithmetic sequence. In the arithmetic sequence, we can find the missing term by taking the difference of two consecutive terms.