Question

Find next in sequence 1,11,21,1211,111221,322211,133221...

Answer 1

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$\underline{\underline{\huge{\textbf{Question:}}}}$

1, 11, 21, 1211, 111221, 312211, 13112221, ......

$\underline{\underline{\huge{\textbf{Solution:}}}}$

I term : 1
II term : 11
III term : 21
IV term : 1211
V term : 111221
VI term : 312211
VII term : 13112221
VIII term : ?

$\underline{\huge{\textsf{Logic :}}}$

To obtain the $n^{th}$ term of the given sequence,
No. of times of occurrence of each digit in the $(n-1)^{th}$ term is placed to the left of each digit of $(n-1)^{th}$ term

$\underline{\large{\textbf{Explaination:}}}$

I term = 1 $\textsf{(One 1)}$

II term = 11 $\textsf{(Two 1's)}$

III term = 21 $\textsf{(One 2 One 1)}$

IV term : 1211 $\textsf{(One 1 One 2 Two 1)}$

V term : 111221 $\textsf{(Three 1's Two 2's One 1)}$

VI term : 312211 $\textsf{(One 3 one 1 Two 2's Two 1's)}$

VII term : 13112221 $\textsf{(One 3 Two 1's Three 2's One 1)}$

VIII term : $\textsf{One 3 Two 1's Three 2's One \implies 13213211}$

$\therefore$ $\textsf{Next term in the Given series is \underline{\textit{13213211}}}$

Thus,
The series is :
$\textsf{1, 11, 21, 1211, 111221, 312211, 13112221,\underline{\textit{13213211}}}$

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