# EXERCISE 19.2

QUESTION 1

Find :

(i) 10 th term of the A.P.

Sol :

We have

a = 1

d = 4 – 1 = 3

= 28

(ii) 18 th term of the A.P.

Sol :

We have :

(iii) n th term of the A.P.

Sol :

We have :

a = 13

d = 8 – 13 = -5

QUESTION 2

If the sequence is an A.P. show that

Sol :

Let the sequence be an A.P. with the first term being A and the common difference being D .
To prove:

ii)

From (i) and (ii), we get:

LHS = RHS

Hence proved

QUESTION 3

(i) Which term of the A.P. is 248 ?

Sol :

a = 3

d = (8-3) = 5

Let

Hence, 248 is the 50 th term of the given A.P.

(ii) Which term of the A.P. is 0 ?

Sol :

a = 84

d = (80-84) = -4

Let

Hence , 0 is the 22nd term of given A.P

(iii) Which term of the A.P. is 254 ?

Sol :

Here , we have

a = 4

d = (9 – 4) = 5

Let

Hence, 254 is the 51 st term of the given A.P.

QUESTION 4

(i) Is 68 a term of the A.P. ?

Sol :

We have

a = 7

d= (10 – 7) = 3

Let

Since n is not a natural number.So, 68 is not a term of the given A.P.

(ii) Is 302 a term of the A.P. ?

Sol :

We have

a= 3

d = (8 – 3) = 5

Let

Since n is not a natural number. So ,302 is not a term of the given A.P.

QUESTION 5

(i) Which term of the sequence is the first negative term?

Sol :

This is an A.P

Here , we have

a = 24

Let the first negative term be

Then , we have :

Thus, the 34 th term is the first negative term of the given A.P.

(ii) Which term of the sequence is (a) purely real (b) purely imaginary?

Sol :

This is an A.P

Here , we have

Let the real term be

Let nth term be purely real

n = 5

So, 5 th term is purely real.

Let n th term be purely imaginary. Then,

So, 13 th term is purely imaginary.

REASON :complex number is a number which can be represented in the form of ‘z=a+ib’ , where a & b are real number and i is an imaginary number, (which is a square root of -1). If a=0 then z is purely imaginary number but if b=0 then z is purely real number.

QUESTION 6

(i) How many terms are there in the A.P.

Sol :

Here , we have

a = 7

d = (10 – 7) =3

Let there be n terms in the given A.P.

Thus, there are 13 terms in the given A.P.

(ii) How many terms are there in the A.P. ?

Sol :

Here , we have

a = -1

Let there be n terms in the given A.P.

Thus, there are 27 terms in the given A.P.

QUESTION 7

The first term of an A.P. is 5 , the common difference is 3 and the last term is 80 ; find the number of
terms.

Sol :

a=5

d=3

a_{n}=80a_{n}=a+(n-1) d\Rightarrow 80=5+(n-1) 3\Rightarrow 75=(n-1) 3\Rightarrow 25=(n-1)\Rightarrow 26=n

*** QuickLaTeX cannot compile formula:

Thus , there are 26 terms in the given A.P

<strong><span style="background-color: #ffff00;">QUESTION 8</span></strong>
The 6 th and 17 th terms of an A.P. are 19 and 41 respectively, find the 40 th term.
Sol :
Given :

*** Error message:
Missing $inserted.  a_{6}=19\Rightarrow a+(6-1) d=19\left[a_{n}=a+(n-1) d\right]\Rightarrow a+5 d=19 \quad \ldots(1)a_{17}=41\Rightarrow a+(17-1) d=41 \quad\left[a_{n}=a+(n-1) d\right]\Rightarrow a+16 d=41 \quad \ldots(2)16 d-5 d=41-19\Rightarrow 11 d=22\Rightarrow d=2d=2a+5 \times 2=19\Rightarrow a=19-10\Rightarrow a=9 *** QuickLaTeX cannot compile formula: We know: *** Error message: Missing$ inserted.



a_{40}=a+(40-1) d \quad\left[a_{n}=a+(n-1) d\right]=a+39 d=9+39 \times 2

*** QuickLaTeX cannot compile formula:

=9+78 =87

<strong><span style="background-color: #ffff00;">QUESTION 9</span></strong>
If 9 th term of an A.P. is zero, prove that its 29 th term is double the 19 th term.
Sol :
To prove:

*** Error message:
Missing $inserted.  a_{29}=2 a_{19} *** QuickLaTeX cannot compile formula: Given : *** Error message: Missing$ inserted.



a_{9}=0\Rightarrow a+(9-1) d=0 \quad\left[a_{n}=a+(n-1) d\right]\Rightarrow a+8 d=0\Rightarrow a=-8 d \ldots(\mathrm{i})

*** QuickLaTeX cannot compile formula:

Proof:
LHS:

*** Error message:
Missing $inserted.  a_{19}=a+(19-1) d=a+18 d= -8d+18d= 10d \dots (ii) *** QuickLaTeX cannot compile formula: RHS : *** Error message: Missing$ inserted.



a_{29}=a+(29-1) d=a+28 d=-8 d+28 d = 20 d \dots (iii)

*** QuickLaTeX cannot compile formula:

Hence proved

<strong><span style="background-color: #ffff00;">QUESTION 10</span></strong>
If 10 times the 10 th term of an A.P. is equal to 15 times the 15 th term, show that 25 th term of the A.P. is zero.
Sol :
Given :

*** Error message:
Missing $inserted.  10 a_{10}=15 a_{15}\Rightarrow 10[a+(10-1) d]=15[a+(15-1) d]\Rightarrow 10(a+9 d)=15(a+14 d)\Rightarrow 10 a+90 d=15 a+210 d\Rightarrow 0=5 a+120 d\Rightarrow 0=a+24 d\Rightarrow a=-24 d \ldots(\mathrm{i}) *** QuickLaTeX cannot compile formula: To show: *** Error message: Missing$ inserted.



a_{25}=0

*** QuickLaTeX cannot compile formula:

Proof :

*** Error message:
Missing $inserted.  \Rightarrow \mathrm{LHS} : a_{25}=a+(25-1) d=a+24 d=-24 d+24 d=0=\mathrm{RHS} *** QuickLaTeX cannot compile formula: Hence , proved <strong><span style="background-color: #ffff00;">QUESTION 11</span></strong> The 10 th and 18 th terms of an A.P. are 41 and 73 respectively. Find 26 th term. Sol : Given : *** Error message: Missing$ inserted.



a_{10}=41\Rightarrow a+(10-1) d=41 \left[a_{n}=a+(n-1) d\right]\Rightarrow a+9 d=41a_{18}=73\Rightarrow a+(18-1) d=73\left[a_{n}=a+(n-1) d\right]\Rightarrow a+17 d=73\Rightarrow 17 d-9 d=73-41\Rightarrow 8 d=32\Rightarrow d=4 \quad \ldots(i)a+9 \times 4=41\Rightarrow a+36=41\Rightarrow a=5 \quad \ldots(ii)

*** QuickLaTeX cannot compile formula:

*** Error message:
Missing $inserted.  a_{26}=a+(26-1) d \quad\left[a_{n}=a+(n-1) d\right]\Rightarrow a_{26}=a+25 d\Rightarrow a_{26}=5+25 \times 4from~(i)~and~(ii)\Rightarrow a_{26}=5+100=105 *** QuickLaTeX cannot compile formula: <strong><span style="background-color: #ffff00;">QUESTION 12</span></strong> In a certain A.P. the 24 th term is twice the 10 th term. Prove that the 72 nd term is twice the 34 th term. Sol : Given : *** Error message: Missing$ inserted.



a_{24}=2 a_{10}\Rightarrow a+(24-1) d=2[a+(10-1) d]\Rightarrow a+23 d=2(a+9 d)\Rightarrow a+23 d=2 a+18 d\Rightarrow 5 d=a \quad \ldots(\mathrm{i})

*** QuickLaTeX cannot compile formula:

To prove:

*** Error message:
Missing $inserted.  a_{72}=2 a_{34} *** QuickLaTeX cannot compile formula: Proof : LHS *** Error message: Missing$ inserted.



: a_{72}=a+(72-1) d\Rightarrow a_{72}=a+71 d\Rightarrow a_{72}=5 d+71 d\Rightarrow a_{72}=76 d

*** QuickLaTeX cannot compile formula:

*** Error message:
Missing $inserted.  \mathrm{RHS} : 2 a_{34}=2[a+(34-1) d]\Rightarrow 2 a_{34}=2(a+33 d)\Rightarrow 2 a_{34}=2(5 d+33 d)\Rightarrow 2 a_{34}=2(38 d)\Rightarrow 2 a_{34}=76 d\therefore \mathrm{RHS}=\mathrm{LHS} *** QuickLaTeX cannot compile formula: Hence, proved. <strong><span style="background-color: #ffff00;">QUESTION 13</span></strong> If *** Error message: Missing$ inserted.



(m+1)(n+1)(3 m+1)(m+n+1)\$ th term.

Sol :

Given :