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Hint: Apply the concepts and formulae of the Arithmetic progression in order to find the common difference of both the sequences provided in the question. Then we will apply the nth term formula of an AP, to find the largest common term of both the sequences.

Let us start by considering the given sequence, 1,11,21,31 ….. upto(100) terms and 31,36,41,46, ….. upto(100) terms.

Let us apply the formula, $d = {a_2} - {a_1}$, where ${a_1}$ is the first term, ${a_2}$ is the second term and d is the common difference.

$

d = {a_2} - {a_1} \\

= 11 - 1 \\

= 10 \\

$

Thus, the common difference of the sequence , 1,11,21,31 ….. upto(100) terms is 10.

Similarly we will find the common difference of the sequence, 31,36,41,46, ….. upto(100).

$

d = {a_2} - {a_1} \\

= 36 - 31 \\

= 5 \\

$

Thus, we get that the common difference of the first and second sequence is 10 and 5, respectively.

Now, we observed that every odd term of the second AP is same as the 1st AP. Also we see that the number of terms in both the given APs is the same. Hence, it can be said that the largest term can be easily decided by the ${n^{th}}$ term of the second AP.

Also, observing again both the given sequences, it can be seen that the first AP contains all the odd terms of the second AP, where the last odd term of 100 terms of an AP is the 99th term.

Thus, we can now apply the nth term formula of an AP, which is given by;

${a_n} = a + \left( {n - 1} \right)d$

Substitute all the known values into the formula to get the largest common term in the given sequences.

$

\Rightarrow {a_n} = 31 + \left( {99 - 1} \right)5 \\

{a_n} = 31 + 98 \times 5 \\

{a_n} = 521 \\

$

Hence, the largest term common in both the sequences 1,11,21,31 ….. upto(100) terms and 31,36,41,46, ….. upto(100) terms is 521, which is option (D).

Note: Always try to form a link or a connection between the given sequence, when it has been asked to find the largest or the smallest term in both the sequences. While forming a link, make sure to not miss any term or element and the sequence obtained after combining the two given sequences must have the common difference, and also you need to be sure that no new term or the element is present in the new sequence, apart from the elements or the terms in the given two sequences.

__Complete step-by-step solution__Let us start by considering the given sequence, 1,11,21,31 ….. upto(100) terms and 31,36,41,46, ….. upto(100) terms.

Let us apply the formula, $d = {a_2} - {a_1}$, where ${a_1}$ is the first term, ${a_2}$ is the second term and d is the common difference.

$

d = {a_2} - {a_1} \\

= 11 - 1 \\

= 10 \\

$

Thus, the common difference of the sequence , 1,11,21,31 ….. upto(100) terms is 10.

Similarly we will find the common difference of the sequence, 31,36,41,46, ….. upto(100).

$

d = {a_2} - {a_1} \\

= 36 - 31 \\

= 5 \\

$

Thus, we get that the common difference of the first and second sequence is 10 and 5, respectively.

Now, we observed that every odd term of the second AP is same as the 1st AP. Also we see that the number of terms in both the given APs is the same. Hence, it can be said that the largest term can be easily decided by the ${n^{th}}$ term of the second AP.

Also, observing again both the given sequences, it can be seen that the first AP contains all the odd terms of the second AP, where the last odd term of 100 terms of an AP is the 99th term.

Thus, we can now apply the nth term formula of an AP, which is given by;

${a_n} = a + \left( {n - 1} \right)d$

Substitute all the known values into the formula to get the largest common term in the given sequences.

$

\Rightarrow {a_n} = 31 + \left( {99 - 1} \right)5 \\

{a_n} = 31 + 98 \times 5 \\

{a_n} = 521 \\

$

Hence, the largest term common in both the sequences 1,11,21,31 ….. upto(100) terms and 31,36,41,46, ….. upto(100) terms is 521, which is option (D).

Note: Always try to form a link or a connection between the given sequence, when it has been asked to find the largest or the smallest term in both the sequences. While forming a link, make sure to not miss any term or element and the sequence obtained after combining the two given sequences must have the common difference, and also you need to be sure that no new term or the element is present in the new sequence, apart from the elements or the terms in the given two sequences.

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