**Q.1: Find the 6th term from the end of the AP 17, 14, 11, …….., (-40) .**

**Sol:**

Here a = 17, d = (14 – 17) = – 3, I = -40

And n = 6

Now, \( n^ {th} \)

= [- 40 – (6 – 1) (-3)]

= [- 40 + 5 x 3]

= – 40+15

= – 25

Hence, the 6th term from the end is – 25.

**Q2: Is 184 a term of the AP 3, 7, 11, 15, ……. ?**

**Sol:**

The given AP is 3, 7, 11, 15, …….

Here, a = 3 and d = 7 – 3 = 4

Let nth term of the given AP be 184.

Then,

\(a_{n} = 184\)

\(\Rightarrow 3 + (n – 1) \times 4 = 184\)

\(\Rightarrow 4n – 1 = 184\)

\(\Rightarrow 4n = 185 \)

\(\Rightarrow n = \frac{184}{4} = 46 \frac{1}{4}\)

But the number of terms cannot be a fraction.

Hence, 184 is not a term of the given AP.

**Q.3: Is – 150 a term of the AP 11, 8, 5, 2, …… ?**

**Sol:**

The given AP is 11, 8, 5, 2, ……

Here, a = 11 and d = 8 – 11 = – 3

Let nth term of the given AP be 150.

Then,

\(a_{n} = 150 \)

\(\Rightarrow 11 + (n – 1) \times (-3) = 150 \)

\(\Rightarrow – 3n + 14 = – 150 \)

\(\Rightarrow – 3n = – 164\)

\(\Rightarrow n = \frac{164}{3} = 54 \frac{2}{3}\)

But the number of terms cannot be a fraction.

Hence, -150 is not a term of the given AP.

**Q.4: Which term of the AP 121, 117, 113, ….. is its first negative term?**

**Sol:**

The given AP is 121, 117, 113, …..

Here, a = 121 and d= 117 – 121 = -4

Let the nth term of the given AP be the first negative term. Then,

\(a_{n} < 0\)

\(\Rightarrow 121 + (n – 1) \times (-4) < 0\)

\(\Rightarrow 125 – 4n < 0\)

\(\Rightarrow -4n < -125\)

\(\Rightarrow n > \frac{125}{4} = 31\frac{1}{4}\)

\(Therefore, n = 32\)

Hence, the 32nd term is the first negative term of the given AP.

**Q.5: Which term of the AP is \(20, 19\frac{1}{4}, 18\frac{1}{2}, 17\frac{3}{4}, …….\) is its negative term?**

**Sol:**

The given AP is \(20, 19\frac{1}{4}, 18\frac{1}{2}, 17\frac{3}{4}, …….\)

Here, a = 20 and d = \( 19\frac{1}{4} – 20 = \frac{77}{4} – 20 = -\frac{3}{4} \)

Let the nth term of the given AP be the first negative term. Then,

\(a_{n} < 0\)

\(\Rightarrow 20 + (n – 1) \times \left ( -\frac{3}{4} \right ) < 0\)

\(\Rightarrow 20 + \frac{3}{4} – \frac{3}{4}n < 0\)

\(\Rightarrow \frac{83}{4} – \frac{3}{4}n < 0 \)

\(\Rightarrow -\frac{3}{4}n < – \frac{83}{4} \)

\(\Rightarrow n > \frac{83}{3} = 27 \frac{2}{3}\)

\(Therefore, n = 28 \)

Hence, the 28th term is the first negative term of the given AP.

**Q.6: The 7th term of an AP is -4 and its 13th term is -16. Find the AP.**

**Sol:**

In the given AP, let the first term = a, and common difference = d

Then, \( T_ {n} \)

\( \Rightarrow \)

\( \Rightarrow \)

Now, \( T_ {7} \)

\( \Rightarrow \)

\( T_{13} \)

\( \Rightarrow \)

Subtracting (1) from (2), we get

\( \Rightarrow \)

Putting d = -2 in (1), we get

a + 6 (-2) = -4

\( \Rightarrow \)

\( \Rightarrow \)

Thus, a = 8, and d = -2

So the required AP is 8, 6, 4, 2, 0 …

**Q.7: The 4 ^{th} term of an AP is 0. Prove that its 25^{th} term is triple its 11^{th} term.**

**Sol:**

In the given AP, let the first term = a, and common difference = d

Then, \( T_ {n} \)

Now, \( T_ {4} \)

\( \Rightarrow \)

\( \Rightarrow \)

Again, \( T_ {11} \)

= -3d + 10 d = 7d [Using (i) ]

Also, \( T_ {25} \)

i.e. \( T_{25} = 3 \times 7d = 3 \times T_{11} \)

Hence, 25th term is triple its 11th term.

**Q.8: The 8 ^{th} term of an AP is zero. Prove that its 38^{th} term is triple its 18^{th} term.**

**Sol:**

Let a be the first term and d be the common difference of the AP. Then,

\(a_{8} = 0\)

\(\Rightarrow a + (8 – 1)d = 0\)

\(\Rightarrow a + 7 d = 0\)

\(\Rightarrow a = -7d \)

Now,

\(\Rightarrow \frac{a_{38}}{a_{18}} = \frac{a + (38 – 1)d}{a + (18 – 1)d}\)

\(\Rightarrow \frac{a_{38}}{a_{18}} = \frac{ – 7d + 37 d} {- 7d + 17d}\)

\(\Rightarrow \frac{a_{38}}{a_{18}} = \frac{30d } {10 d} = 3\)

\(a_{38} = 3 \times a_{18}\)

Hence, the 38th term of the AP is triple its 18th term.

**Q.9: The 4 ^{th} term of an AP is 11. The sum of the 5^{th} and 7^{th} term of this AP is 34. Find its common difference.**

**Sol:**

Let a be the first term and d be the common difference of the AP. Then,

\(a_{4} = 11\)

\(\Rightarrow a + (4 – 1)d = 11\)

\(\Rightarrow a + 3 d = 11\)

Now,

\(a_{5} + a_{7} = 34\)

\(\Rightarrow (a + 4d) + (a + 6d) = 34\)

\(\Rightarrow 2a + 10d = 34\)

\(\Rightarrow a + 5d = 17\)

From (i) and (ii) we get

11 – 3d + 5d = 17

\(\Rightarrow 2d = 17 – 11 = 6\)

\(\Rightarrow d = 3\)

Hence, the common difference of the AP is 3.

**Q.10: The 9 ^{th} term of an AP is -32 and the sum of its 11^{th} and 13^{th} term is -94. Find the common difference of the AP.**

**Sol:**

Let a be the first term and d be the common difference of the AP. Then,

\(a_{4} = 11\)

\(\Rightarrow a + (9 – 1)d = – 32 \)

\(\Rightarrow a + 8 d = -32 \)

Now,

\(a_{11} + a_{13} = -94 \)

\(\Rightarrow (a + 10d) + (a + 12d) = -94 \)

\(\Rightarrow 2a + 22d = -94 \)

\(\Rightarrow a + 11d = -47\)

From (i) and (ii) we get

– 32 – 8d + 11d = -47

\(\Rightarrow 3d = -47 + 32 = -15 \)

\(\Rightarrow d = -5 \)

Hence, the common difference of the AP is -5.

**Q.11: Determine the nth term of the AP whose 7 ^{th} term is -1 and 16^{th} term is 17.**

**Sol: **

Let a be the first term and d be the common difference of the AP. Then,

\(a_{ 7 }\)

→ a + (7 – 1) d = -1 \(\left [ a_{ n } = a + (n – 1)d \right ]\)

→ a + 6d = -1 … (1)

Also,

\(a_{ 16 }\)

→ a + 15d = 17 …(2)

From (1) and (2), we get

-1 – 6d + 15d = 17

→ 9d = 17 + 1 = 18

→ d = 2

Putting d = 2 in (1), we get

a + 6 x 2 = -1

→ a = -1 – 12 = -13

Therefore, \(a_{ n } = a + (n – 1)d\)

= -13 + (n – 1) x 2

= 2n – 15

Hence, the nth term of the AP is (2n – 15)

**Q.12: If 4 times the 4 ^{th} term of an AP is equal to 18 times its 18^{th} term then find its 22^{nd} term.**

**Sol:**

Let a be the first term and d be the common difference of the AP. Then,

4 x \(a_{ 4 } = 18 \times a_{ 18 }\)

→ 4 (a + 3d) = 18 (a + 17d) \(\left [ a_{ n } = a + (n – 1)d \right ]\)

→ 2 (a + 3d) = 9 (a + 17d)

→ 2a + 6d = 9a + 153d

→ 7a = -147d

→ a = -21d

→ a + 21d = 0

→ a + (22 – 1)d = 0

→ \(a_{ 22 }\)

Hence, the 22^{nd} term of the AP is 0.

**Q.13: If 10 times the 10 ^{th} term of an AP is equal to 15 times the 15^{th} term, show that its 25^{th} term is zero.**

**Sol:**

Let a be the first term and d be the common difference of the AP. Then,

10 x \(a_{ 10 } = 15 \times a_{ 15 }\)

→ 10 (a + 9d) = 15 (a + 14d) \(\left [ a_{ n } = a + (n – 1)d \right ]\)

→ 2 (a + 9d) = 3(a + 14d)

→ 2a + 18d = 3a + 42d

→ a = -24d

→ a + 24d = 0

→ a + (25 – 1) d = 0

→ \(a_{ 25 }\)

Hence, the 25^{th} term of the AP is 0.

**Q.14: Find the common difference of an AP whose first term is 5 and the sum of its first four terms is half the sum of the next four terms.**

**Sol:**

Let the common difference of the AP be d.

First term, a = 5

Now,

\(a_{ 1 } + a_{ 2 } + a_{ 3 } + a_{ 4 } = \frac{ 1 }{ 2 } (a_{ 5 } + a_{ 6 } + a_{ 7 } + a_{ 8 })\)

→ a + (a + d) + (a + 2d) + (a + 3d) = \(\frac{ 1 }{ 2 }\)

** **\(\left [ a_{ n } = a + (n – 1)d \right ]\)

→ 4a + 6d = \(\frac{ 1 }{ 2 }\)

→ 8a + 12d = 4a + 22d

→ 22d – 12d = 8a – 4a

→ 10d = 4a

→ d = \(\frac{ 2 }{ 5 }a\)

→ d = \(\frac{ 2 }{ 5 } \times 5 = 2\)

Hence, the common difference of the AP is 2.

**Q.15: The sum of the 2 ^{nd} and the 7^{th} terms of an AP is 30. If its 15^{th} term is 1 less than twice its 8^{th} term, find the AP.**

**Sol:**

Let a be the first term and d be the common difference of the AP. Then,

\(a_{ 2 } + a_{ 7 } = 30\)

Therefore, (a + d) + (a + 6d) = 30 \(\left [ a_{ n } = a + (n – 1)d \right ]\)

→ 2a + 7d = 30 ….(1)

Also,

\(a_{ 15 } = 2a_{ 8 } – 1\)

→ a + 14d = 2 (a + 7d) – 1

→ a + 14d = 2a + 14d – 1

→ -a = -1

→ a = 1

Putting a = 1 in (1), we get

2 x 1 + 7d = 30

→ 7d = 30 – 2 = 28

→ d = 4

So,

\(a_{ 2 }\)

\(a_{ 3 }\)

Hence, the AP is 1, 5, 9, 13,…

**Q.16: For what value of n, the nth terms of the arithmetic progressions 63, 65, 67,… and 3, 10, 17,.. are equal?**

**Sol:**

Let the nth term of the given progressions be \(t_{ n }\)

The first AP is 63, 65, 67,…

Let its first term be a and the common difference be d.

Then a = 63, and d = (65 – 63) = 2

So, its \(n^{th}\)

\(T_{ n }\)

→ 3 + (n -1) x 7

→ 7n – 4

Now, \(t_{ n }\)

→ 61 + 2n = 7n – 4

→ 65 = 5n

→ n = 13

Hence, the 13^{th} terms of the AP’s are the same.

**Q.17: The 17 ^{th} term of an AP is 5 times more than twice its 8^{th} term. If the 11^{th }term of the AP is 43, find its nth term.**

**Sol:**

Let a be the first term and d be the common difference of the AP. Then,

\(a_{ 17 } = 2a_{ 8 } + 5\)

Therefore, a + 16d = 2 (a + 7d) + 5 \(\left [ a_{ n } = a + (n – 1)d \right ]\)

→ a + 16d = 2a + 14d + 5

→ a – 2d = -5 …(1)

Also,

\(a_{ 11 }\)

→ a + 10d = 43 …(2)

From (1) and (2), we get

-5 + 2d + 10d = 43

→ 12d = 43 + 5 = 48

→ d = 4

Putting d = 4 in (1), we get

a – 2 x 4 = -5

→ d = 4

Therefore, \(a_{ n } = a + (n – 1)d\)

= 3 + (n – 1) x 4

= 4n – 1

Hence, the nth term of the AP is (4n – 1)