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

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.