Question

Choose the correct option provided below for the following question.
Find the odd one among: \[97,77,59,43,26,17\].
A) \[77\]
B) \[59\]
C) \[43\]
D) \[26\]

Answer 1

Hint: Here if we observe carefully, we have six numbers in the series which are all close to the perfect squares. We have to add or subtract a certain number so we can get the perfect squares. Here the added or subtracted numbers are also in an arithmetic progression.

Complete step-by-step solution:
It is the given terms, \[97,77,59,43,26,17\].
Now let us add or subtract the least possible number to each of the terms to make them close to their nearest squares.
If we add \[3\] to \[97\], we get the nearest perfect square \[100\].
\[97 + 3 = 100\]
If we add \[4\] to \[77\], we get the nearest perfect square \[81\].
\[77 + 4 = 81\]
If we add \[5\] to \[59\], we get the nearest perfect square \[64\].
\[59 + 5 = 64\]
If we add \[6\] to \[43\], we get the nearest perfect square \[49\].
\[43 + 6 = 49\]
If we subtract \[1\] from \[26\], we get the nearest perfect square \[25\].
\[26 - 1 = 25\]
If we subtract \[1\] from \[17\], we get the nearest perfect square \[16\].
\[17 - 1 = 16\]
The added or subtracted numbers to get a perfect square are as follows:
\[3,4,5,6,1,1\]
In the above sequence every consecutive number is increasing by \[1\] until the last two numbers that are in an arithmetic progression, which decreased and then remained the same. So, in this series, the first four numbers are directly in progression while the last two show no such relation. Therefore, \[26,17\] are the odd ones in the given series.
Since there is no option given for \[17\], the correct answer is \[26\] from the given options.

$\therefore $ The correct option is D.

Note: From the solution of the question, we can observe that, in mathematics, a sequence is a list of objects (or events) which have been ordered in a sequential fashion such that each member either comes before or after every other member. More formally, a sequence is a function with a domain equal to the set of positive integers. A series is a sum of a sequence of terms. That is, a series is a list of numbers with additional operations between them.