Arithmetic Sequence: Inserting Five Arithmetic Means

Introduction to Arithmetic Means

• The video introduces the concept of arithmetic means, which are numbers that lie between two extremes in an arithmetic sequence.

• The specific problem presented is to insert five arithmetic means between the numbers 18 and 54.

Understanding the Problem

• The extremes of the sequence are identified as 18 (first term) and 54 (last term). The goal is to find five terms in between these values.

• Each term in the sequence is denoted as a_1, a_2, ldots, a_7 , where a_1 = 18 and a_7 = 54 . The missing terms are a_2, a_3, a_4, a_5, and a_6 .

Formula for Finding Terms

• To find the missing terms, the formula used is:

[

a_n = a_1 + d(n - 1)

]

where:

• a_n is the last term (54),

• a_1 is the first term (18),

• d is the common difference,

• n represents the number of terms (7).

Solving for Common Difference

• Substituting known values into the formula gives:

[

54 = 18 + d(7 - 1)

]

which simplifies to:

[

54 = 18 + 6d

]

This leads to solving for d:

[

36 = 6d

]

Thus,

[

d = 36/6 = 6

].

Finding Missing Terms

• With d = 6, we can now calculate each missing term:

• Second Term:

a_2 = a_1 + d = 18 + 6 = 24.

• Third Term:

a_3 = a_2 + d = 24 + 6 = 30.

• Fourth Term:

a_4 = a_3 + d =30 +6=36.

• Fifth Term:

a_5=a_4+d=36+6=42.

• Sixth Term:

Finally,

a_6=42+6=48. Thus all five arithmetic means have been found as follows: [24,30,36,42,and48].

Conclusion