Answer

Verified

433.5k+ views

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. \\

\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.

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. \\

\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.

Recently Updated Pages

Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred

The branch of science which deals with nature and natural class 10 physics CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

What is BLO What is the full form of BLO class 8 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE