Manipur Board Solutions for Chapter: Arithmetic Progression (AP), Exercise 1: EXERCISE 6.1

In an $\mathrm{A}.\mathrm{P}$, the $5^{\text{th }}$ term is $7$ and the ${11}^{\text{th }}$ term is $10$. Find the first term and the common difference.

The $6^{\text{th }}$ and the ${17}^{\text{th }}$ terms of an $\mathrm{A}.\mathrm{P}$ are $21$ and $54$ respectively. Find the ${40}^{\text{th }}$ term.

The ${15}^{\text{th }}$ term of an $\mathrm{AP}$ exceeds the ${20}^{\text{th }}$ term by $8$. Find the common difference.

How many two-digit numbers are divisible by $3$?

If the ${\mathrm{m}}^{\text{th }}$ term of an $\mathrm{AP}$ be $\frac{1}{\mathrm{n}}$ and the ${\mathrm{n}}^{\text{th }}$ term be $\frac{1}{ \mathrm{m}}$, then show that the $(\mathrm{mn}{)}^{\mathrm{th}}$ term is $1$.

If $a,b,c$ be the $p^{th},q^{\text{th }}$ and $r^{\text{th }}$ terms of an $\mathrm{AP}$, prove that $a(q-r)+b(r-p)+c(p-q)=0$.

A sum of $₹5000$ is invested at $6\%$ per annum simple interest. Calculate the interest at the end of each year and show that they form an $\mathrm{AP}$. Also, find the interest at the end of the ${25}^{\text{th }}$ year.

In an auditorium, the seats are so arranged that there are $8$ seats in the first row, $11$ seats in the second, $14$ seats in the third etc. thereby increasing the number of seats by$3$  every next row. If there are $50$ seats in the last row, how many rows of seats are there in the auditorium?