Step 1
Let \(A_1, A_2, A_3, A_4, A_5\) be five arithmetic means between 5 and 21.
Then \(5, A_1, A_2, A_3, A_4, A_5, 21\) are in arithmetic sequence with first term and seventh term are
\(a_1 = 5,\)

\(a_7 = 21\) Step 2 Now nth term of arithmetic sequence is \(a_n = a_1 + (n - 1)d\) For \(n=7,\) \(a_7 - a_1 + (7 - 1)d\)

\(21 = 5 + 6d\)

\(6d = 21 - 5\)

\(6d = 16\) \(d = \frac{7}{3}\) Step 3 Therefore, the five arithmetic means between 5 and 21 are \(A_1 = a_1 + d = 5 + \frac{8}{3} = \frac{23}{3}\)

\(A_2 = a_1 + 2d = 5 + 2 \times \frac{8}{3} = \frac{31}{3}\)

\(A_3 = a_1 + 3d = 5 + 3 \times \frac{8}{3} = \frac{39}{3} = 13\)

\(A_4 = a_1 + 4d = 5 + 4 \times \frac{8}{3} = \frac{47}{3}\)

\(A_5 = a_1 + 5d = 5 + 5 \times \frac{8}{3} = \frac{55}{3}\) Step 4 Ans:The five arithmetic means between 5 and 21 are \(\frac{23}{3}, \frac{31}{3}, 13, \frac{47}{3}, \frac{55}{3}\)

\(a_7 = 21\) Step 2 Now nth term of arithmetic sequence is \(a_n = a_1 + (n - 1)d\) For \(n=7,\) \(a_7 - a_1 + (7 - 1)d\)

\(21 = 5 + 6d\)

\(6d = 21 - 5\)

\(6d = 16\) \(d = \frac{7}{3}\) Step 3 Therefore, the five arithmetic means between 5 and 21 are \(A_1 = a_1 + d = 5 + \frac{8}{3} = \frac{23}{3}\)

\(A_2 = a_1 + 2d = 5 + 2 \times \frac{8}{3} = \frac{31}{3}\)

\(A_3 = a_1 + 3d = 5 + 3 \times \frac{8}{3} = \frac{39}{3} = 13\)

\(A_4 = a_1 + 4d = 5 + 4 \times \frac{8}{3} = \frac{47}{3}\)

\(A_5 = a_1 + 5d = 5 + 5 \times \frac{8}{3} = \frac{55}{3}\) Step 4 Ans:The five arithmetic means between 5 and 21 are \(\frac{23}{3}, \frac{31}{3}, 13, \frac{47}{3}, \frac{55}{3}\)