Question

Which term of the AP: $3,\text{ }8,\text{ }13,\text{ }18,\text{ }.\text{ }.\text{ }.\text{ },\text{ }is\text{ }78$?

Answer 1

In order to find solution to this Arithmetic Progression Problem, we have to use a formula for finding the $n-th$ term of an Arithmetic Progression that is ${{a}_{n}}=a+\left( n-1 \right)d$ , where ${{a}_{n}}$ is the value of the ${{n}^{th}}$ term, $a$ is the initial term, $n$ is the total number of terms and $d$ is the common difference, to find which term in this Arithmetic Series is $78$.

Complete step by step solution:
We have our given series as $3,\text{ }8,\text{ }13,\text{ }18,\text{ }.\text{ }.\text{ }.\text{ },78$.
With this we have to find which term is $78$.
Therefore, we will apply a formula for finding the ${{n}^{th}}$ term of an Arithmetic Progression that is ${{a}_{n}}=a+\left( n-1 \right)d$ .
With this, we get:
Last term in an Arithmetic series, ${{a}_{n}}=78$
First term, $a=3$
Common difference, $d={{a}_{2}}-{{a}_{1}}=8-3=5$
${{n}^{th}}$ term we have to find, $n=?$
Since we have all we got to evaluate into the formula, therefore, putting values into these formulas, we get:
${{a}_{n}}=a+\left( n-1 \right)d$
On evaluating, we get:
$\Rightarrow 78=3+\left( n-1 \right)\times 5$
On taking $+3$ on left-hand side, we get our expression as:
$\Rightarrow 78-3=\left( n-1 \right)\times 5$
On simplifying, we get:
$\Rightarrow 75=\left( n-1 \right)\times 5$
Now, on taking $5$ on left-hand side and applying sign rule, we get our expression as:
$\Rightarrow \dfrac{75}{5}=\left( n-1 \right)$
On simplifying, we get:
$\Rightarrow 15=\left( n-1 \right)$
On further simplification and eliminating brackets, we get our expression as:
$\Rightarrow 15=n-1$
Now, on taking $-1$ on left-hand side and applying sign rule, we get:
$\Rightarrow 15+1=n$
On Simplifying, we get:
$\Rightarrow n=16$

Therefore, the ${{16}^{th}}$ term of an Arithmetic Progression is $78$.

Note: Arithmetic Progression (AP) is a sequence of numbers in order in which the difference of any two consecutive numbers is a constant value.
We have two major formulas which is related to ${{n}^{th}}$ term of Arithmetic Progression:
To find the ${{n}^{th}}$ term of A.P: ${{a}_{n}}=a+\left( n-1 \right)d$
To find sum of ${{n}^{th}}$ term of A.P: $S=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)$
Based on the given question of an Arithmetic Progression, we have to decide which formula we have to use.