Question

The ratio of the second and seventh of an arithmetic mean between -7 and 65 is 1 by 7 find n?

Answer 1

$\text{Let A_1,A_2,......A_n be the n arithmetic means between -7 and 65}$

$\text{Then,}$

$A_1=-7+d$

$A_2=-7+2d$

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.

.

$A_n=-7+nd$

$\textbf{Given:}$

$A_2:A_7=1:7$

$\textbf{To find:}$

$\text{The value of n}$

$\textbf{Solution:}$

$\text{Consider,}$

$A_2:A_7=1:7$

$\dfrac{A_2}{A_7}=\dfrac{1}{7}$

$\dfrac{-7+2d}{-7+7d}=\dfrac{1}{7}$

$7(-7+2d)=-7+7d$

$-49+14d=-7+7d$

$7d=42$

$\implies\,d=\dfrac{42}{7}$

$\implies\boxed{\bf\,d=6}$

$\text{Then,}$

$A_n=59$

$-7+nd=59$

$-7+n(6)=59$

$n(6)=66$

$\implies\,n=\dfrac{66}{6}$

$\implies\boxed{\bf\,n=11}$