Question

Find ${{t}_{n}}$ for the A.P. $3$, $8$, $13$, $18$,…..

Answer 1

Hint: For this problem we need to calculate the ${{n}^{th}}$ term of the given A.P series which is represented by ${{t}_{n}}$. Consider the given series of A.P and compare the terms with ${{t}_{0}}$, ${{t}_{1}}$, ${{t}_{2}}$, ${{t}_{3}}$, ${{t}_{4}}$,….. Calculate the common difference of the A.P which is equal to the difference between any two consecutive numbers and represents with $d$. Now use the formula for the ${{n}^{th}}$ term of the A.P by using the formula ${{t}_{n}}={{t}_{0}}+\left( n-1 \right)d$. Substitute all the value we have in the above equation and simplify it by using the mathematical operations.

Complete step by step solution:
Given a series of A.P of $3$, $8$, $13$, $18$,…..
Comparing the given terms in the A.P with the terms ${{t}_{0}}$, ${{t}_{1}}$, ${{t}_{2}}$, ${{t}_{3}}$, ${{t}_{4}}$,….., then we will get
${{t}_{0}}=3$, ${{t}_{1}}=8$, ${{t}_{2}}=13$, ${{t}_{3}}=18$.
Calculating the common difference of the A.P by calculating the difference between the any two consecutive terms of the A.P.
$d={{t}_{1}}-{{t}_{0}}$
Substituting the values of ${{t}_{1}}$ and ${{t}_{0}}$in the above equation, then we will get
$\begin{align}
  & d=8-3 \\
 & \Rightarrow d=5 \\
\end{align}$
We have the formula for the ${{n}^{th}}$ term of the A.P as ${{t}_{n}}={{t}_{0}}+\left( n-1 \right)d$. Substituting all the values we have in the above formula, then we will get
${{t}_{n}}=3+\left( n-1 \right)5$
Apply distribution law of multiplication over subtraction in the above equation, then we will have
$\begin{align}
  & {{t}_{n}}=3+5n-5 \\
 & \Rightarrow {{t}_{n}}=5n-2 \\
\end{align}$

Hence the ${{n}^{th}}$ of the given A.P series $3$, $8$, $13$, $18$,….. is ${{t}_{n}}=5n-2$.

Note: In this problem they have calculated the formula for the ${{n}^{th}}$ term. We can also check whether the calculated result is correct or not by substituting $n=1,2,3,...$ and compare those values with the values of given ${{t}_{1}}$, ${{t}_{2}}$, ${{t}_{3}}$, ${{t}_{4}}$. If we get both the values as same, then our calculated result is correct.