Ex.5.1 Q4 Arithmetic progressions Solutions - NCERT Maths Class 10

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Question

Which of the followings are APs? If they form an AP, Find the common difference d and write three more terms.

i) \(\begin{align}2,4,16\dots\dots\end{align}\)

ii)\(\begin{align}2,\frac{5}{2},3,\frac{7}{2}\dots\dots\end{align}\)

iii) \(\begin{align}- 1.2, - 3.2, - 5.2, - 7.2\dots\dots\end{align}\)

iv)\(\begin{align}{\kern 1pt} \,\,\, - 10, - 6, - 2,2\dots\dots\end{align}\)

v)\(\begin{align}3,3 + \sqrt 2 ,\,3 + 2\sqrt 2 ,3 + 3\sqrt 2 \dots\dots\end{align}\)

vi) \(\begin{align}0.2,0.22,0.222,0.2222\dots\dots\end{align}\)

vii)\(\begin{align}0, - 4, - 8, - 12\dots\dots\end{align}\)

viii)\(\begin{align}- \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}\dots\dots\end{align}\)

ix) \(\begin{align}1,3,9,27\dots\dots\end{align}\)

x)\(\begin{align}a,2a,3a.4a,\dots\dots\end{align}\)

xi) \(\begin{align}a,{a^2},{a^3},{a^4}\end{align}\)

xii) \(\begin{align}\sqrt 2 ,\sqrt 8 ,\sqrt {18} ,\sqrt {132} \end{align}\)

xiii)\(\begin{align}\sqrt 3 ,\sqrt 6 ,\sqrt 9 ,\sqrt {12} \end{align}\)

xiv)\(\begin{align}{1^2},{3^2},{5^2},{7^2}\end{align}\)

xv)\(\begin{align}{1^2},{3^2},{5^2},{7^3}\end{align}\)

 Video Solution
Arithmetic Progressions
Ex 5.1 | Question 4

Text Solution

 

 Reasoning:

General form of an arithmetic progression is \(\begin{align} a, (a+d), (a+2d), (a+3d)…\end{align}\) where \(a\) is the first term and \(d\) is the common difference.

i) What is Known?

\(\begin{align} 2, 4, 8, 16…..\end{align}\)

What is Unknown?

Common difference and next three more terms of AP if it is an AP.

Steps:

The given numbers are \(2,\,4,\,8,\,16\)

First term \( a =2\)

Common difference

 \(d=a_{2}-a_{1}=4-2=2\)

Common difference

 \(d=a_{3}-a_{2}=8-4=4\)

\(\begin{align}({a_{3}-a_{2}) \neq (a_{2}-a_{1})}\end{align}\)

\(\begin{align}2, 4, 8, 16\end{align}\) are not in AP, because the common difference is not equal.

ii) What is Known?

\(\begin{align}2,\,\frac{5}{2},\,3,\,\frac{7}{2}\end{align}\)

What is Unknown?

Common difference and next three more terms of AP if it is an AP.

Steps:

The given numbers are \(\begin{align}2,\,\frac{5}{2},\,3,\,\frac{7}{2}\end{align}\)

First term a \(= 2\)

Common difference  
\[\begin{align}& d =a_{2}-a_{1}=\frac{5}{2}-2=\frac{5-4}{2}=\frac{1}{2}\end{align}\]

Common difference  
 \[\begin{align}& d =a_{3}-a_{2}=3-\frac{5}{2}=\frac{6-5}{2}=\frac{1}{2}\end{align}\]

Since 

\(\begin{align}({a_3} - {a_2}) = ({a_2} - {a_1}).\end{align}\)

\(\begin{align}2,\,\frac{5}{2},\,3,\,\frac{7}{2}\end{align}\) forms an AP and common difference is\(\begin{align}\frac{1}{2}\end{align}\) .

The next three terms are:

Fifth term

\[\begin{align} &=a+4 d \\ &=2+4 \times \frac{1}{2} \\ &=2+2 \\ &=4 \end{align}\]

Sixth term

\[\begin{align} &=a+5 d \\ &=2+5 \times \frac{1}{2} \\ &=2+\frac{5}{2} \\ &=\frac{4+5}{2} \\ &=\frac{9}{2} \end{align}\]

Seventh term

\[\begin{align} &=a+6 d \\ &=2+6 \times \frac{1}{2} \\ &=5 \end{align}\]

\(\begin{align}2,\,\frac{5}{2},\,3,\,\frac{7}{2}\end{align}\) forms an AP and the common difference is \(\begin{align}\frac{1}{2}\end{align}\). The next three terms are \(\begin{align}4,\,\frac{9}{2}\,,5.\end{align}\)

iii) What is Known?

\(\begin{align} - 1.2,\, - 3.2,\, - 5.2,\, - 7.2 \ldots \ldots .\end{align}\)

What is Unknown?

Whether it forms an AP. If it is find the common difference and the next three terms of AP.

Steps:

The given numbers are

\(\begin{align} - 1.2,\, - 3.2,\, - 5.2,\, - 7.2 \ldots \ldots .\end{align}\)

First term \(\begin{align} a =  -1.2\end{align}\)

Common difference  

\[\begin{align} d &={a_2} - {a_1}\\& = - 3.2 - ( - 1.2)\\& = - 3.2 + 1.2 = - 2\end{align}\]

Common difference

\[\begin{align} &={a_3} - {a_2} = - 5.2 - ( - 3.2)\\ &= - 5.2 + 3.2 = - 2 \end{align}\]

Since 

\(\begin{align}({a_3} - {a_2}) = ({a_2} - {a_1}) . \end{align}\)

It forms an AP.

The fifth term

\[\begin{align} &=a+4d \\ &=-1.2+4(-2) \\ &=-1.2-8=-9.2 \end{align}\]

The sixth term

\[\begin{align} &=a+5d \\ &=-1.2+5(-2) \\ &=-1.2-10 \\ &=-11.2 \end{align}\]

The seventh term 

\[\begin{align} &=a+6d \\ &=-1.2+6(-2) \\ &=-1.2-12 \\ &=-13.2 \end{align}\]

\(\begin{align} - 1.2,\, - 3.2,\, - 5.2,\, - 7.2\end{align}\) forms an AP with common difference -2. The next three terms of AP are \(\begin{align}-9.2, -11.2, -13.2. \end{align}\)

iv) What is Known?

\(\begin{align} -10, -6, -2, 2 \end{align}\)

What is Unknown?

Whether it forms an AP. If it is find the common difference and the next three terms of AP.

Steps:

The given numbers are \(\begin{align} -10, -6, -2, 2 \end{align}\)

First term a \(= -10\) 

Common difference  

\[\begin{align} d &=a_{2}-a_{1} \\ &=-6-(-10) \\ &=-6-10 \\ &=4 \end{align}\]

Common difference 

\[\begin{align} d &=a_{3}-a_{2} \\ &=-2-(-6) \\ &=-2+6 \\ &=4 \end{align}\]

Since

\((a_{3}-a_{2})=(a_{2}-a_{1})\)

Fifth Term: \( a+4 d=-10+16=6\)

Sixth Term:  \(a+5 d=-10+20=10\)

Seventh Term:  \(a+6 d=-10+24=14\)

\(\begin{align}-10, -6 ,-2, 2 \end{align}\) forms an AP with common difference \(4\) and next terms are \(6, 10, 14.\)

v) What is Known?

\(\begin{align}3,\,\,3 + \sqrt 2 \,\,,3 + 2\sqrt 2 \,\,,3 + 3\sqrt 2 \ldots \ldots \end{align}\)

What is Unknown?

Given list of numbers form an AP or not. If it is find the common difference and the next three terms of AP.

Steps:

The list of numbers is

\(\begin{align}3,\,\,3 + \sqrt 2 \,\,,3 + 2\sqrt 2 \,\,,3 + 3\sqrt 2 \ldots \ldots \end{align}\)

Common difference

\[\begin{align} d   &=a_{2}-a_{1} \\ &=3+\sqrt{2}-3 \\ &=\sqrt{2} \end{align}\]

Common difference

\[\begin{align} d  &=a_{3}-a_{2} \\ &=3+2 \sqrt{2}-(3+\sqrt{2}) \\ &=3+2 \sqrt{2}-3-\sqrt{2} \\ &=\sqrt{2} \end{align}\]

Since

\(\begin{align} a_{3}-a_{2}=a_{2}-a_{1}\end{align}\)

So \(\begin{align}3,\,\,3 + \sqrt 2 \,\,,3 + 2\sqrt 2 \,\,,3 + 3\sqrt 2 \ldots \ldots \end{align}\) forms an AP with common difference 4.

Next three terms are

\(\begin{align} \text { Fifth term } &=a+4 d \\ &=3+4 \sqrt{2} \\ \text { Sixth term } &=a+5 d \\ &=3+5 \sqrt{2} \\ \text { Seventh term } &=a+6 d \\ &=3+6 \sqrt{2} \end{align}\)

It is an AP with common difference \(\begin{align}\sqrt 2 \end{align}\) and Next three terms are

\(\begin{align}3 + 4\sqrt 2 ,\,\,3 + 5\sqrt 2 \,\,,3 + 6\sqrt 2 \end{align}\) ,.

vi) What is Known?

\(\begin{align}0.2,0.22,0.222,0.2222.....\end{align}\)

What is Unknown?

Given list of numbers form an AP or not. If it is find the common difference and the next three terms of AP.

Steps:

The list of numbers are

\(\begin{align}0.2,0.22,0.222,0.2222.....\end{align}\)

Common difference 
\[\begin{align} d &=a_{2}-a_{1} \\ &=0.22-0.2 \\ &=0.20 \end{align}\]

Common difference
\[\begin{align} d &=a_{3}-a_{2} \\ &=0.222-0.220 \\ &=0.002\\ (a_{3}-a_{2} )&\neq (a_{2}-a_{1})\end{align}\]

The given list of numbers does not form an AP.

vii) What is Known?

\(\begin{align} 0,-4,-8,-12……\end{align}\)

What is Unknown?

Given list of numbers form an AP or not. If it is find the common difference and the next three terms of AP.

Steps:

The list of numbers is

\(\begin{align} 0,-4,-8,-12……\end{align}\)

Common difference 
\(\begin{align}d = {a_2} - {a_1}= - 4 – 0 = -4\end{align}\)

Common difference 
\(\begin{align}d = {a_3} - {a_2}= -8 – (-4) = -8 + 4 = -4\end{align}\)

Since\(\begin{align}({a_3} - {a_2}) = ({a_2} - {a_1})\end{align}\) . It forms an AP.

\(\begin{align} \text { Fifth term } &=a+4 d \\ &=0+4(-4) \\ &=-16 \\ \text { Sixth term } &=a+5 d \\ &=0+5(-4) \\ &=-20 \\ \text { Seventh term } &=a+6 d \\ &=0+6(-4) \\ &=-24 \end{align}\)

The given numbers form an AP with difference \(-4.\) The next three terms are \(\begin{align} -16, -20, -24. \end{align}\)

viii) What is Known?

\(\begin{align} - \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}.......\end{align}\)

What is Unknown?

Given list of numbers form an AP or not. If it is find the common difference and the next three terms of AP.

Steps:

The given list of numbers is

\(\begin{align} - \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}.......\end{align}\)

Common difference 
\[\begin{align} d &= {a_2} - {a_1}\\&= - \frac{1}{2} - \left( { - \frac{1}{2}} \right)\\ &= - \frac{1}{2} + \frac{1}{2}\\&= 0\end{align}\]

Common difference
\[\begin{align} d &= {a_3} - {a_2}\\& = - \frac{1}{2} - \left( { - \frac{1}{2}} \right)\\ &= - \frac{1}{2} + \frac{1}{2}\\&= 0\end{align}\]

Since \(\begin{align}({a_3} - {a_2}) = ({a_2} - {a_1})\end{align}\) .The list of numbers forms an AP.

\(\begin{align}{\text{The fifth term }} &= a + 4d\\& = - \frac{1}{2} + 4(0)\\ &= - \frac{1}{2}\\{\text{ The sixth term }} &= a + 5d\\ &= - \frac{1}{2} + 5(0)\\ &= - \frac{1}{2}\\{\text{The seventh term}} &= {a} + 6d\\ &= - \frac{1}{2} + 6(0)\\ &= - \frac{1}{2}\end{align}\)

The given list of numbers form an AP with common difference \(d = 0.\) Next three terms are \(\begin{align} - \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}.\end{align}\)

ix) What is Known?

\(\begin{align}1, 3 ,9 ,27.\end{align}\)

What is Unknown?

Given list of numbers form an AP or not. If it is find the common difference and the next three terms of AP.

Steps:

The list of numbers are \(\begin{align}1, 3 ,9 ,27.\end{align}\)

Common difference \(\begin{align} d = {a_2} - {a_1}= 3 – 1 = 2\end{align}\)

Common difference \(\begin{align} d = {a_3} - {a_2}= 9 -3 =6\end{align}\)

Since\(\begin{align} ({a_2} - {a_1}) \ne ({a_3} - {a_2})\end{align}\)

The given list of numbers does not form an AP.

x) What is Known?

\(\begin{align}{a, 2a, 3a, 4a}\end{align}\)

What is Unknown?

Given list of numbers form an AP or not. If it is find the common difference and the next three terms of AP.

Steps:

The list of numbers is \(\begin{align}{a, 2a, 3a, 4a}.\end{align}\)

Common difference
\[\begin{align} d  &=a_{2}-a_{1} \\ &=2a-a\\ &=a \end{align}\]


Common difference 
\[\begin{align} d &=a_{3}-a_{2} \\ &=3a-2a \\ &=a \end{align}\]

since\(\begin{align}{a_3} - {a_2} = {a_2} - {a_1}, {a, 2a, 3a, 4a}\end{align}\)

 forms an AP.

The fifth term \(=a+4d=a+4a=5a\)

The sixth term \(=a+5d=a+5a=6a\) 

The seventh term \(=a+6d=a+6a=7a\)

The given list of numbers form an AP with common difference \(d =a\) The next three terms are \(5a, 6a, 7a.\)

xi) What is Known?

\(\begin{align}a,{a^2},{a^3},{a^4}\end{align}\)

What is Unknown?

Given list of numbers form an AP or not. If it is find the common difference and the next three terms of AP.

Steps:

The list of numbers is \(\begin{align}a,{a^2},{a^3},{a^4}\end{align}\)

Common difference 

\[\begin{align} d &=a_{2}-a_{1} \\ &=a^{2}-a \\ &=a(a-1) \end{align}\]

Common difference 

\[\begin{align} d &= {a_3} - {a_2}\\ &= {a^3} - {a^2}\\ &= {a^2}\left( {a - 1} \right)\end{align}\]

Since 

\(\begin{align}({a_2} - {a_1}) \ne ({a_3} - {a_2})\end{align}\)

The given list of numbers does not form an AP.

xii) What is Known?

\(\begin{align}\sqrt 2 ,\sqrt 8 ,\sqrt {18} ,\sqrt {32} \end{align}\)

What is Unknown?

Given list of numbers form an AP or not. If it is find the common difference and the next three terms of AP.

Steps:

The list of numbers is \(\begin{align}\sqrt 2 ,\sqrt 8 ,\sqrt {18} ,\sqrt {32} \end{align}\)

Common difference 

\[\begin{align}d &={a_2} - {a_1}\\&= \sqrt 8 - \sqrt 2 \\ &= \sqrt {4 \times 2} - \sqrt 2 \\ &= 2\sqrt 2 - \sqrt 2 \\ &= \sqrt 2 \end{align}\]

Common difference

\[\begin{align}d &= {a_3} - {a_2}\\ &= \sqrt {18} - \sqrt 8 \\ &= \sqrt {9 \times 2} - \sqrt {4 \times 2} \\ &= 3\sqrt 2 - 2\sqrt 2 \\& = \sqrt 2 \end{align}\]

since

\(\begin{align}({a_2} - {a_1}) = ({a_3} - {a_2})\end{align}\) The given numbers form an AP.

\(\begin{align}{\text{The fifth term }} &= a + 4d\\ &= \sqrt 2 + 4\sqrt 2 \\ &= 5\sqrt 2 \\ &= \sqrt {25 \times 2} \\ &= \sqrt {50} \\{\text{The sixth term}}& = a + 5d\\ &= \sqrt 2 + 5\sqrt 2 \\ &= 6\sqrt 2 \\ &= \sqrt {36 \times 2} \\ &= \sqrt {72} \\{\text{ The seventh term }} &= {\rm{a}} + 6d\\ &= \sqrt 2 + 6\sqrt 2 \\ &= 7\sqrt 2 \\ &= \sqrt {49 \times 2} \\ &= \sqrt {98} \end{align}\)

The list of numbers forms an AP with common difference\(\begin{align}\sqrt 2 \end{align}\). Next three terms are \(\begin{align}\sqrt {50} ,\sqrt {72} ,\sqrt {98} \end{align}\)

xiii ) What is Known?

\(\begin{align}\sqrt 3 ,\,\sqrt 6 \,,\sqrt 9 ,\,\sqrt {12} \end{align}\)

What is Unknown?

Given list of numbers form an AP or not. If it is find the common difference and the next three terms of AP.

Steps:

The list of numbers is

\(\begin{align}\sqrt 3 ,\,\sqrt 6 \,,\sqrt 9 ,\,\sqrt {12} \end{align}\)

Common difference 

\[\begin{align}d &= {a_2} - {a_1} = \sqrt 6 - \sqrt 3 \\ &= \sqrt {3 \times 2} - \sqrt 3 \\ &= \sqrt 3 \left( {\sqrt 2 - 1} \right)\end{align}\]

Common difference 

\[\begin{align} d&= {a_3} - {a_2} = \sqrt 9 - \sqrt 6 \\ &= \sqrt {3 \times 3} - \sqrt {3 \times 2} \\ &= \sqrt 3 (\sqrt 3 - \sqrt 2 )\end{align}\]

since\(({a_2} - {a_1}) \ne ({a_3} - {a_2})\)

The given list of numbers does not form an AP.

xiv) What is Known?

\(\begin{align}{1^2},\,{3^2}\,,{5^2}\,,{7^2}\end{align}\)

What is Unknown?

Given list of numbers form an AP or not. If it is find the common difference and the next three terms of AP.

Steps:

The list of numbers is

\(\begin{align}{1^2},\,{3^2}\,,{5^2}\,,{7^2}\end{align}\)

Common difference 

\(\begin{align} d= {a_2} - {a_1}= 25 – 1 = 24\end{align}\)

Common difference

\( d ={a_3} - {a_2}= 49 – 25 = 24\)

Since \(({a_2}- {a_1}) = ({a_3} - {a_2})\) they form an AP

\(\begin{align} \text { The fifth term } &=a+4 d \\ &=1+4 \times 24 \\ &=1+96 \\ &=97 \end{align}\)

\(\begin{align} \text { The sixth term } &=a+5 d \\ &=1+5 \times 24 \\ &=1+120 \\ &=121 \end{align}\)

\(\begin{align} \text { The seventh term } &=a+6 d \\ &=1+6 \times 24 \\ &=1+144 \\ &=145 \end{align}\)

The list of numbers form an AP with common difference \(24.\) The next three terms are \(97, 121,\) and \(145.\)

xv)What is Known?

\(\begin{align}{1^2},\,{3^2},\,{5^2},\;{7^3}\end{align}\)

What is Unknown?

Given list of numbers form an AP or not. If it is find the common difference and the next three terms of AP.

Steps:

The list of numbers

\(\begin{align}{1^2},\,{3^2},\,{5^2},\;{7^3}\end{align}\)

Common difference 

\[\begin{align} d = {a_2} - {a_1}= 9 – 1 = 8\end{align}\]

Common difference

\[\begin{align}d = {a_3} - {a_2}= 25 - 9 = 16\end{align}\]

Since

\(\begin{align}({a_2} - {a_1}) \ne ({a_3} - {a_2})\end{align}\)

The given list of numbers does not form an AP.

  
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