# How do you find the arithmetic means of the sequence 3, __, __, __, __, __, 27?

Feb 26, 2017

$\text{The Reqd. Means are, } 7 , 11 , 15 , 19 \mathmr{and} 23.$

#### Explanation:

Let the reqd. $5$ Arithmetic Means be ${A}_{1} , {A}_{2} , {A}_{3} , {A}_{4} , {A}_{5.}$

Then, $3 , {A}_{1} , {A}_{2} , {A}_{3} , {A}_{4} , {A}_{5} , 27$ are $7$ terms of an A.P., for

which, in the Usual Notation, $a = 3 \mathmr{and} {a}_{7} = 27.$

But, for an A.P., ${a}_{n} = a + \left(n - 1\right) d , n \in \mathbb{N} .$

$\therefore a = 3 , {a}_{7} = 27 \Rightarrow 27 = 3 + \left(7 - 1\right) d \Rightarrow 24 = 6 d \Rightarrow d = 4.$

Hence, ${A}_{1} = {a}_{2} = a + d = 3 = 4 = 7 , {A}_{2} = {a}_{3} = {a}_{2} + 4 = 11 ,$

${A}_{3} = {A}_{2} + 4 = 15 , {A}_{4} = 19 , {A}_{5} = 23.$

Thus, the Reqd. Means are, $7 , 11 , 15 , 19 \mathmr{and} 23.$

Enjoy Maths.!