Question

If be an arithmetic mean between two numbers and $S$ be the sum of $n$ arithmetic means between the same numbers, then.

• A. $S = n A$
• B. $A = n S$
• C. $A = S$
• D. None of these

Answer 1

Answer: (A)

$S = n A$

Answer 2

Let the two quantities be $a$ and $b$ and let be the $n$ A.M.'s between them. Then $a , A _ { 1 } , A _ { 2 } \ldots \ldots . A _ { n } , b$ are in A.P. and let $d$ be the common difference.

Now $T _ { n + 2 } = b = a + ( n + 2 - 1 ) d \Rightarrow d = \frac { b - a } { n + 1 }$

Also $A _ { 1 } + A _ { 2 } + \ldots \ldots + A _ { n } = S _ { n + 1 } - a$

$= \frac { 1 } { 2 } ( n + 1 ) \left[ 2 a + ( n + 1 - 1 ) \frac { ( b - a ) } { ( n + 1 ) } \right] - a$

=$\frac { n } { 2 } [ 2 a + ( b - a ) ] = \frac { n } { 2 } ( a + b ) = n \left( \frac { a + b } { 2 } \right) = n A$.

Trick: Let 1,3,5,7,9 is in A.P. In this series

$A = 5 , n = 3 , S = 15$ ⇒ $S = n A$ .