Question

Which term of the A.P: $3,8,13,18,....\text{Is }78?$

Answer 1

First assume 78 as some nth term. Next we can find the common difference by subtracting two consecutive terms. Later we can use the nth term formula and we can find the value of n.
nth term formula:
${{a}_{n}}=a+(n-1)d$
Where, ${{a}_{n}}$ is the nth term, $a$ is the first term and $d$ is a common difference.

Complete step-by-step answer:
In this question we have to find the value of n in the above formula.
Given,
${{a}_{n}}=78$ here we assumed that $78$ is the nth term.
$a=3$ , First term.
$d=5$
We got a common difference by subtracting the first and second term.
$\begin{aligned} & d=8-3 \\\ & d=5 \\\ \end{aligned}$
Now, putting these values in the above formula, we get:
$\begin{aligned} & 78=3+(n-1)5 \\\ & 75=(n-1)5 \\\ & (n-1)=15 \\\ & n=16 \\\ \end{aligned}$

Hence, we got the value of $n=16.$

Additional Information:
In mathematics we have 3 types of progressions. Geometric progression, Arithmetic progression and harmonic progression. Out of these three the arithmetic progression is the simplest one which uses linear term and common difference. In geometric progression we have a common ratio and harmonic progression is reciprocal of arithmetic progression. These progressions have a great application in many fields from both technical and non-technical. And they also served as a mathematical tool to deal with finite series.

Note: A.P stands for Arithmetic progression. In this progression we have a first term and every subsequent term has a common difference. So we can use this fact that every term is at a fixed value from the first term and that fixed value is a multiple of common term.
We can verify the above answer by putting the value of n.
$\begin{aligned} & {{a}_{n}}=3+(16-1)5 \\\ & {{a}_{n}}=3+75 \\\ & {{a}_{n}}=78 \\\ \end{aligned}$