Answer
355.3k+ views
Hint: We have given n arithmetic means between 3 and 17 that means we have an AP of n+2 terms where first term is 3 and last term is 17. And use the data given in question to proceed further.
Complete step-by-step answer:
Let ‘n’ A.M.s be ${A_1},{A_2},{A_3}, \ldots ,{A_n}$
So we have the given A.P is
$3,{A_1},{A_2}, \ldots ,{A_n},17$
In our case we have
${a_n} = 17,a = 3$ and number of terms = (n+2)
$\because {a_n} = a + \left( {n - 1} \right)d$
$
17 = 3 + \left[ {\left( {n + 2} \right) - 1} \right]d \\
17 - 3 = \left( {n + 1} \right)d \\
\therefore d = \dfrac{{14}}{{\left( {n + 1} \right)}} \\
$
Now
$
{A_1} = a + d = 3 + \dfrac{{14}}{{n + 1}} \\
{A_n} = a + nd = 3 + n\dfrac{{14}}{{n + 1}} \\
$
As given in question,
$
\dfrac{{{A_n}}}{{{A_1}}} = \dfrac{3}{1} \\
\Rightarrow \dfrac{{3 + \dfrac{{14n}}{{n + 1}}}}{{3 + \dfrac{{14}}{{n + 1}}}} = 3 \\
$
$
\Rightarrow \dfrac{{3\left( {n + 1} \right) + 14n}}{{3\left( {n + 1} \right) + 14}} = 3 \\
\Rightarrow \dfrac{{17n + 3}}{{3n + 17}} = 3 \\
\Rightarrow 17n + 3 = 3\left( {3n + 17} \right). \\
\Rightarrow 17n + 3 = 9n + 51 \\
\Rightarrow 8n = 48 \Rightarrow n = 6 \\ $
Note: Whenever you get this type of question the key concept of solving is you have to solve like an AP of n+2 terms and simple mathematics to use the data given in question and use the formula of finding the last term of AP to get an answer.
Complete step-by-step answer:
Let ‘n’ A.M.s be ${A_1},{A_2},{A_3}, \ldots ,{A_n}$
So we have the given A.P is
$3,{A_1},{A_2}, \ldots ,{A_n},17$
In our case we have
${a_n} = 17,a = 3$ and number of terms = (n+2)
$\because {a_n} = a + \left( {n - 1} \right)d$
$
17 = 3 + \left[ {\left( {n + 2} \right) - 1} \right]d \\
17 - 3 = \left( {n + 1} \right)d \\
\therefore d = \dfrac{{14}}{{\left( {n + 1} \right)}} \\
$
Now
$
{A_1} = a + d = 3 + \dfrac{{14}}{{n + 1}} \\
{A_n} = a + nd = 3 + n\dfrac{{14}}{{n + 1}} \\
$
As given in question,
$
\dfrac{{{A_n}}}{{{A_1}}} = \dfrac{3}{1} \\
\Rightarrow \dfrac{{3 + \dfrac{{14n}}{{n + 1}}}}{{3 + \dfrac{{14}}{{n + 1}}}} = 3 \\
$
$
\Rightarrow \dfrac{{3\left( {n + 1} \right) + 14n}}{{3\left( {n + 1} \right) + 14}} = 3 \\
\Rightarrow \dfrac{{17n + 3}}{{3n + 17}} = 3 \\
\Rightarrow 17n + 3 = 3\left( {3n + 17} \right). \\
\Rightarrow 17n + 3 = 9n + 51 \\
\Rightarrow 8n = 48 \Rightarrow n = 6 \\ $
Note: Whenever you get this type of question the key concept of solving is you have to solve like an AP of n+2 terms and simple mathematics to use the data given in question and use the formula of finding the last term of AP to get an answer.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)