Answer

Verified

340.8k+ views

**Hint:**This problem is from arithmetic progression. We are given three terms in the form of n and they are in A.P. so it is clear that there is a common difference d in two consecutive terms. So we will form two equations in n and d form. One equation from first and second term and the other is from first and third term. On solving them we will get the value of n and d. from that we will find the numbers. So let’s solve it!

**Step by step solution:**

Given that \[\left( {2n - 1} \right)\] , \[\left( {3n + 2} \right)\] and \[\left( {6n - 1} \right)\] are in A.P.

So let d be the common difference. Then we can write,

\[\left( {3n + 2} \right) - \left( {2n - 1} \right) = d\] ……..equation1

Now on solving the equation above we get,

\[ \Rightarrow 3n + 2 - 2n + 1 = d\]

Taking similar terms on one side we get,

\[ \Rightarrow 3n - 2n + 2 + 1 = d\]

\[ \Rightarrow n + 3 = d\] …….equation1.1

Now the difference in first and third term is 2d. then we can write,

\[\left( {6n - 1} \right) - \left( {2n - 1} \right) = 2d\] …….equation2

Now on solving the equation above we get,

\[ \Rightarrow 6n - 1 - 2n + 1 = 2d\]

Taking similar terms on one side we get,

\[ \Rightarrow 6n - 2n - 1 + 1 = 2d\]

\[ \Rightarrow 4n = 2d\]

So on further simplification,

\[ \Rightarrow n = \dfrac{d}{2}\]

\[d = 2n\]

This is the value of d.

From 1.1 we get \[ \Rightarrow d - n = 3\]

Putting the value of d in the equation above,

\[ \Rightarrow 2n - n = 3\]

\[ \Rightarrow n = 3\]

This is the value of n \[ \Rightarrow n = 3\] .

Now putting the value one by one in the numbers given we get the numbers also.

First number: \[\left( {2n - 1} \right) = 2 \times 3 - 1 = 6 - 1 = 5\]

Second number: \[\left( {3n + 2} \right) = 3 \times 3 + 2 = 9 + 2 = 11\]

Third number: \[\left( {6n - 1} \right) = 6 \times 3 - 1 = 18 - 1 = 17\]

**Thus the numbers are \[5,11,17\].**

**Note:**

Note that the numbers are given that they are already in A.P. so they have the relation in them. We can solve the problem by finding the value of common difference also. That is just need to find the first number and rest two can be found by adding the common difference. How? See below.

\[

\Rightarrow n + 3 = d \\

\Rightarrow d - n = 3 \\

\Rightarrow d - \dfrac{d}{2} = 3 \\

\Rightarrow \dfrac{d}{2} = 3 \\

\Rightarrow d = 3 \times 2 = 6 \\

\]

Now First number: \[\left( {2n - 1} \right) = 2 \times 3 - 1 = 6 - 1 = 5\]

Then second number \[5 + d = 5 + 6 = 11\]

Then third number \[11 + d = 11 + 6 = 17\]

Recently Updated Pages

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

What happens when entropy reaches maximum class 11 chemistry JEE_Main

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Why is the adrenaline hormone called fight or flight class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between lanthanoids and actinoids class 12 chemistry CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Give 10 examples of unisexual and bisexual flowers

Open circulatory system is present in I Arthropods class 12 biology CBSE

Name the highest peak of the Indian Himalayas class 8 social science CBSE