     Question Answers

# There are ‘n’ A.M.s between 3 and 17. The ratio of the last mean to the first mean is 3:1 .find the value of n.  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.

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.
View Notes
What are the Factorsof 17?  Factors of 17  Table of 17 - Multiplication Table of 17  What are the Domains of the Earth  Ratio And Proportion  What are the Challenges of Democracy?  What are the Functions of the Human Skeletal System?  Concept of Ratio  Failures are The Pillars of Success Essay  What are the Successor and Predecessor?  