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

Last updated date: 16th May 2024
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Hint: Find the common difference of the AP first, and then apply the formula for the ${{r}^{th}}$ term of an Arithmetic Progression.

Complete step-by-step answer:
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. \\
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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