# Ex.5.2 Q7 Arithmetic Progressions Solution - NCERT Maths Class 10

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## Question

Find the $$31^\rm{st}$$ term of an AP whose $$11^\rm{th}$$ term is $$38$$ and the $$16^\rm{th}$$ term is $$73.$$

Video Solution
Arithmetic Progressions
Ex 5.2 | Question 7

## Text Solution

What is Known?

$$11^\rm{th}$$ and $$16^\rm{th}$$ term of AP.

What is Unknown?

$$31^\rm{st}$$ term of AP.

Reasoning:

$${a_n} = a + \left( {n - 1} \right)d$$ is the general term of AP. Where $${a_n}$$ is the $$n\rm{th}$$ term, $$a$$ is the first term, $$d$$ is the common difference and $$n$$ is the number of terms.

Steps:

$\begin{array}{l}{a_n} = a + (n - 1)d\\{a_{11}} = 38\\a + (11 - 1)d = 38\\a + 10d = 38 \qquad \dots\left( 1 \right)\\{a_{16}} = 73\\a + 15d = 73\,\qquad \dots(2)\end{array}$

By solving the two equations (1) & (2) for $$a,d$$

\begin{align}5d &= 35\\d &= 7\end{align}

Putting $$d$$ in the (1) equation

\begin{align}a &= 38 - 70\\ &= - 32\end{align}

$$31^\rm{st}$$ terms is,

\begin{align}{a_{31}} &= a + (31 - 1)d\\ &= - 32 + 30 \times 7\\ &= - 32 + 210\\& = 178\end{align}

The $$31^\rm{st}$$ term of AP is $$178.$$