Question

If \[{t_n}\] denotes \[{n^{th}}\] term of the series \[2 + 3 + 6 + 11 + 18 + \ldots \ldots \], then \[{t_{50}}\] is
(a) \[2 + {49^2}\]
(b) \[2 + {48^2}\]
(c) \[2 + {50^2}\]
 (d) \[2 + {51^2}\]

Answer 1

Hint: Here, we need to find the value of \[{t_{50}}\]. We will use the given information to find the rewrite of the first five terms of the given series. Then, we will use these to form a generalised equation for the \[{n^{th}}\] term of the series. Finally, we will use the generalised equation for \[{n^{th}}\] term of the series to find and simplify the value of \[{t_{50}}\].

Complete step-by-step answer:
First, we will rewrite the first five terms of the given series.
The first term of the series is 2.
The number 2 is the sum of 0 and 2.
Therefore, we get
\[2 = 0 + 2\]
The square of 0 is 0.
Thus, we can rewrite the equation as
\[2 = {0^2} + 2\]
Therefore, we get
First term of the series \[ = {0^2} + 2\]
The second term of the series is 3.
The number 3 is the sum of 1 and 2.
Therefore, we get
\[3 = 1 + 2\]
The square of 1 is 1.
Thus, we can rewrite the equation as
\[3 = {1^2} + 2\]
Therefore, we get
Second term of the series \[ = {1^2} + 2\]
The third term of the series is 6.
The number 6 is the sum of 4 and 2.
Therefore, we get
\[6 = 4 + 2\]
The square of 2 is 4.
Thus, we can rewrite the equation as
\[6 = {2^2} + 2\]
Therefore, we get
Third term of the series \[ = {2^2} + 2\]
The fourth term of the series is 11.
The number 11 is the sum of 9 and 2.
Therefore, we get
\[11 = 9 + 2\]
The square of 3 is 9.
Thus, we can rewrite the equation as
\[11 = {3^2} + 2\]
Therefore, we get
Fourth term of the series \[ = {3^2} + 2\]
The fifth term of the series is 18.
The number 18 is the sum of 16 and 2.
Therefore, we get
\[18 = 16 + 2\]
The square of 4 is 16.
Thus, we can rewrite the equation as
\[18 = {4^2} + 2\]
Therefore, we get
Fifth term of the series \[ = {4^2} + 2\]
It is given that \[{t_n}\] denotes \[{n^{th}}\] term of the series \[2 + 3 + 6 + 11 + 18 + \ldots \ldots \].
Since \[n\] denotes a term of the series, therefore \[n\] has to be a natural number.
Therefore, the first five terms are \[{t_1}\], \[{t_2}\], \[{t_3}\], \[{t_4}\], and \[{t_5}\].
Therefore, we get the equations
\[{t_1} = {0^2} + 2\]
\[{t_2} = {1^2} + 2\]
\[{t_3} = {2^2} + 2\]
\[{t_4} = {3^2} + 2\]
\[{t_5} = {4^2} + 2\]
Rewriting the equations, we get
\[{t_1} = {\left( {1 - 1} \right)^2} + 2\]
\[{t_2} = {\left( {2 - 1} \right)^2} + 2\]
\[{t_3} = {\left( {3 - 1} \right)^2} + 2\]
\[{t_4} = {\left( {4 - 1} \right)^2} + 2\]
\[{t_5} = {\left( {5 - 1} \right)^2} + 2\]
Thus, from the above equations, we can generalise the formula for \[{t_n}\] as
\[{t_n} = {\left( {n - 1} \right)^2} + 2\]
Now, we will find the value of \[{t_{50}}\].
Substituting \[n = 50\] in the generalised equation \[{t_n} = {\left( {n - 1} \right)^2} + 2\], we get
\[ \Rightarrow {t_{50}} = {\left( {50 - 1} \right)^2} + 2\]
Subtracting the terms in the parentheses, we get
\[ \Rightarrow {t_{50}} = {49^2} + 2\]
Therefore, we get the value of \[{t_{50}}\] as \[{49^2} + 2\].
Thus, the correct option is option (a).
Note: We used the term ‘natural numbers’ in the solution. Natural numbers include all the positive integers like 1, 2, 3, etc. For example: 1, 2, 100, 400, 52225, are all natural numbers. The natural numbers are the basic numbers we use when counting objects.
A common mistake is to use whole numbers instead of natural numbers, and write the first term of the series as \[{t_0}\]. This is incorrect because if the first term is taken as \[{t_0}\] instead of \[{t_1}\], then the \[{t_{50}}\] term would denote the 51st term, and not the 50th term.