Question

find 5 arthmatic mean between 5 and 80​

Answer 1

$\large\underline\purple{\bold{Solution :- }}$

Let the 5 arithmetic means be u, v, w, x, y.

So, resulting series

• 5, u, v, w, x, y, 80 is an AP series.

with

• First term, a = 5

• Number of terms, n = 7

• Last term, aₙ = 80

Let the common difference of the series be 'd'.

We know that

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

On substituting the values, we get

$\rm :\implies\:80 \: = \: 5 + (7 - 1)d$

$\rm :\implies\:75 = 6d$

$\rm :\implies\:d \: = \: \dfrac{75}{6}$

$\boxed{ \purple{ \implies \bf \: d \: = \: 12.5}}$

Hence,

• u = 5 + 12.5 = 17.5

• v = 17.5 + 12.5 = 30

• w = 30 + 12.5 = 42.5

• x = 42.5 + 12.5 = 55

• y = 55 + 12.5 = 67.5

Hence,

5 arithmetic means between 5 and 80 are

• 17.5, 30, 42.5, 55, 67.5