find 20 airthmetic means between 4 and 67

Question

vishweshshrimali5 · Answer

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, ...