Question

The \[{100^{th}}\] term of the sequence \[1,{\rm{ }}2,{\rm{ }}2,{\rm{ }}3,{\rm{ }}3,{\rm{ }}3,{\rm{ }}4,{\rm{ }}4,{\rm{ }}4,{\rm{ }}4 \ldots \ldots \] is
A. \[12\]
B. \[13\]
C. \[14\]
D. \[15\]

Answer 1

Hint: In the preceding problem, we are given an endless sequence of natural integers as \[1,2,2,3,3,3,4,4,4,4.....\] where n successive entries have the value n. We must locate the \[{100^{th}}\] phrase in the provided sequence. To reach the solution, we must first identify the structure of each term's last position in this sequence. The sequence's needed term will then be the term with a value at position \[100\]

Formula Used: The last place of natural numbers can be determined using
\[ \Rightarrow \dfrac{{n\left( {n + 1} \right)}}{2}\]

Complete step by step solution: We have been provided in the question that,
The sequence is,
\[1,{\rm{ }}2,{\rm{ }}2,{\rm{ }}3,{\rm{ }}3,{\rm{ }}3,{\rm{ }}4,{\rm{ }}4,{\rm{ }}4,{\rm{ }}4 \ldots \ldots \]
And we are asked to determine the \[{100^{th}}\] term of the sequence.
Now, we have to write the terms from the given sequence, we have
Term one \[ = 1\]
Term two\[ = 2\]
Term four\[ = 3\]
Term seven\[ = 4\]
Term eleven\[ = 5\]
From the above let us consider that,
\[S = 1 + 2 + 4 + 7 + 11...n\;terms\]
Now, using the above expression, we have to write the expression for \[{a_n}\]\[{a_n}\; = {\rm{ }}1{\rm{ }} + {\rm{ }}\left( {1{\rm{ }} + {\rm{ }}2{\rm{ }} + {\rm{ }}3{\rm{ }} + {\rm{ }}..\left( {n - 1} \right){\rm{ }}terms} \right)\]
\[ = {\rm{ }}\dfrac{{1{\rm{ }} + {\rm{ }}n\left( {n - 1} \right)}}{2}\]
Now, we have to solve the numerator of the above equation, we obtain
\[ = \dfrac{{({n^2}-n + 2)}}{2}\]
Now, let us assume
If
\[n = 14\]
Then
\[{a_{14}}\; = \dfrac{{({{14}^2}-14 + 2)}}{2}\]
Now, we have to simplify the power, we get
\[ = {\rm{ }}\dfrac{{\left( {196-14 + 2} \right)}}{2}\]
On simplifying the numerator, we get
\[ = \dfrac{{184}}{2}\]
On simplifying further, we get
\[ = 92\]
Now, it is concluded that 92nd term is \[14\]
Now, let us have
\[n = 15\]
Then,
\[{a_{15}}\; = {\rm{ }}\dfrac{{({{15}^2}\;-{\rm{ }}15{\rm{ }} + {\rm{ }}2)}}{2}\]
Now, we have to solve the power, we get
\[ = {\rm{ }}\dfrac{{(225\;-{\rm{ }}15{\rm{ }} + {\rm{ }}2)}}{2}\]
On simplifying the numerator of the above equation, we get
\[ = \dfrac{{212}}{2}\]
On further simplification, we get
\[{a_{15}}\; = 106\]
It is understood that 106th term is \[15\]
Therefore, 100th term of the sequence will be \[14\]

Option ‘C’ is correct

Note: A sequence is a grouping of a finite or infinite number of things or pieces, or a set of numbers, in a certain order that is maintained by some unique rule. Each element's order, i.e. rank or position, is distinct and special in this case. Implementing the one-of-a-kind rule yields the nth term of the sequence \[{a_n}\].