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# Which term of the AP : $3,8,13,18,...,$ is $78$ ?  Hint: Find the common difference of the AP first, and then apply the formula for the ${{r}^{th}}$ term of an Arithmetic Progression.

An Arithmetic Progression is any series of numbers, in which the successive terms have the same difference amongst them. For example, if we say that four numbers, $a,b,c,d$ are in an AP, or form an Arithmetic Progression, then it means that the differences between the successive terms, i.e. $a$ and $b$, $b$ and $c$, and $c$ and $d$ are all equal to each other.
Written mathematically, this simply means that $b-a=c-b=d-c$.
Now, we can say that the ${{r}^{th}}$ term of an AP can be written as : $a+(r-1)d$, where $a=$ the first term of the AP given, and $d=$ the common difference between the successive terms in the AP.
Thus, in this question, it will be wise to proceed with finding the common difference, or $d$ first. We can pick any two successive terms and do so.
Let’s pick the ${{1}^{st}}$ and ${{2}^{nd}}$ terms of the AP given. Thus, we have the common difference as $8-3=5$. Note that, it’ll be the same if we pick any other pair of consecutive terms. Even if we picked the ${{2}^{nd}}$ and ${{3}^{rd}}$ term, we’d get the common difference $d=13-8=5$. Hence, the common difference of this AP is $d=5.$
Now, we can clearly see that the first term of the AP given is $3$. Thus, $a=3$.
Now, we can simply substitute these values in the formula of the ${{r}^{th}}$ term, we’ll get the index of the term which is equal to $78$.
Therefore, \begin{align} & a+(r-1)d=78 \\ & \Rightarrow 3+(r-1)5=78 \\ & \Rightarrow (r-1)5=75 \\ & \Rightarrow r-1=15 \\ & \Rightarrow r=16. \\ \end{align}
Therefore, we can see that $78$ is the ${{16}^{th}}$ term of the given AP.

Note: Be very cautious while applying the formula, the ${{r}^{th}}$term is given by multiplying $(r-1)$ with the common difference, not $r$. Students tend to mix up the two.
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