Exercise 5.2

Fill in the blanks in the following table, given that $a$ is the first term, $d$ the common difference and $a_n$ the $n$th term of the AP:

$$\begin{array}{|c|c|c|c|}

\hline
a & d & n & a_n \\
\hline
7 & 3 & 8 & \text{.} \\
\hline
-18 & \text{.} & \text{.} & 100 \\
\hline
\text{.} & \text{.} & -3 & 18 \\
\hline
-18.9 & 2.5 & \text{.} & 3.6 \\
\hline
3.5 & 0 & 105 & \text{.} \\
\hline
\end{array}$$

The $17$th term of an AP exceeds its $10$th term by $7$. Find the common difference.

Which term of the AP: $3, 15, 27, 39, \ldots$ will be $132$ more than its $54^{\text{th}}$ term?

Two APs have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms?

How many three-digit numbers are divisible by $7$?

How many multiples of $4$ lie between $10$ and $250$?

For what value of $n$, are the $n$th terms of two APs: $63, 65, 67, \ldots$ and $3, 10, 17, \ldots$ equal?

Determine the AP whose third term is $16$ and the $7$th term exceeds the $5$th term by $12$.

Find the 20th term from the last term of the AP : $3, 8, 13, \ldots, 253$.

The sum of the $4^{\text{th}}$ and $8^{\text{th}}$ terms of an AP is $24$ and the sum of the $6^{\text{th}}$ and $10^{\text{th}}$ terms is $44$. Find the first three terms of the AP.

Subba Rao started work in 1995 at an annual salary of $5000$ and received an increment of $200$ each year. In which year did his income reach $7000$?

Choose the correct choice in the following and justify :

(i) 30th term of the AP: $10, 7, 4, \dots$, is (A) $97$ (B) $77$ (C) $-77$ (D) $-87$

(ii) 11th term of the AP: $-3, \dfrac{1}{2}, 2, \dots$, is (A) $28$ (B) $22$ (C) $-38$ (D) $-48$

Ramkali saved $\text{₹}\ 5$ in the first week of a year and then increased her weekly savings by $\text{₹}\ 1.75$. If in the $n$th week, her weekly savings become $\text{₹}\ 20.75$, find $n$.

In the following APs, find the missing terms in the boxes :

(i) $2, \boxed{\phantom{0}}, 2\sqrt{6}$

(ii) $\boxed{\phantom{0}}, 13, \boxed{\phantom{0}}, 3$

(iii) $5, \boxed{\phantom{0}}, \boxed{\phantom{0}}, 19\sqrt{2}$

(iv) $-4, \boxed{\phantom{0}}, \boxed{\phantom{0}}, \boxed{\phantom{0}}, \boxed{\phantom{0}}, 6$

(v) $\boxed{\phantom{0}}, 38, \boxed{\phantom{0}}, \boxed{\phantom{0}}, \boxed{\phantom{0}}, -2\sqrt{2}$

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

Find the number of terms in each of the following APs : (i) $7, 13, 19, \dots, 205$ (ii) $18, \dfrac{11}{5}, 13, \dots, -47$

Check whether $-150$ is a term of the AP: $11, 8, 5, 2, \ldots$

Find the $31$th term of an AP whose $11$th term is $38$ and the $16$th term is $73$.

An AP consists of $50$ terms of which the $3$rd term is $12$ and the last term is $106$. Find the $29$th term.

If the $3$rd and the $9$th terms of an AP are $4$ and $-8$ respectively, which term of this AP is zero?