Question

find 20 airthmetic means between 4 and 67

Answer 1

Lets assume these arithmetic means to be \(\large{a_1, a_2, a_3,..., a_{20}}\). Then,

\(\large{4, a_1, a_2, ..., a_{20}, 67}\) must form an AP.

Let, the common difference of this AP = d, then,

\(\large{a_1 = 4+d}\)

Similarly, 67 is 22nd term of this AP.

Thus, \[\large{67 = 4 + (22-1)d}\]

\[\large{\implies 63 = 21d}\]

\[\large{\implies d = \cfrac{63}{21}}\]

\[\large{\implies d = 3}\]

Thus, \(\large{a_1 = 4 + d = 4+3 = 7}\)

\(\large{a_2 = 4 + (3-1)d = 4 + 2* d = 4+2*3 = 10}\)

and so on.

Thus, the required arithmetic means = 7, 10, 13, 16, 19, ...