# In N A.M.'S Are Introduced Between 3 and 17 Such that the Ratio of the Last Mean to the First Mean is 3 : 1, Then the Value of N is - Mathematics

MCQ

In n A.M.'s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is

#### Options

• 6

• 8

• 4

•  none of these.

#### Solution

6

Let

$A_1 , A_2 , A_3 , A_4 . . . . A_n$ be the n arithmetic means between 3 and 17.
Let d be the common difference of the A.P. 3,

$A_1 , A_2 , A_3 , A_4 , . . . . A_n$ and 17.
Then, we have:

d = $\frac{17 - 3}{n + 1}$ = $\frac{14}{n + 1}$

Now,

$A_1$ = 3 + d = 3 + $\frac{14}{n + 1}$ = $\frac{3n + 17}{n + 1}$

And,

$A_n = 3 + nd = 3 + n\left( \frac{14}{n + 1} \right) = \frac{17n + 3}{n + 1}$

$\therefore \frac{A_n}{A_1} = \frac{3}{1}$

$\Rightarrow \frac{\left( \frac{17n + 3}{n + 1} \right)}{\left( \frac{3n + 17}{n + 1} \right)} = \frac{3}{1}$

$\Rightarrow \frac{17n + 3}{3n + 17} = \frac{3}{1}$

$\Rightarrow 17n + 3 = 9n + 51$

$\Rightarrow 8n = 48$

$\Rightarrow n = 6$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 19 Arithmetic Progression
Q 5 | Page 51