Interpolation of Arithmetic Means

An arithmetic progression consists of a numerical sequence that respects a general formation condition. It is worth remembering that the numerical logic between the elements of a progression is given by the ratio between the arranged numbers. Interpolating arithmetic means means determining the real numbers existing between the extreme values ​​of a numerical sequence, so as to become an Arithmetic Progression. For this, we need to recall some situations involving PA. Look:

Formula for calculating the General Term

Each sequential term of a PA is dependent on the value of the ratio and the 1st term. Watch:

THE₂ = the₁ + r
THE₃ = the₁ + 2r
THE₄ = the₁ + 3r
THE₅ = the₁ + 4r
THE₆ = the₁ + 5r
THE₇ = the₁ + 6r
THE₈ = the₁ + 7r

And so on.

To determine the elements existing between the extreme values ​​of an AP, we need the value of the ratio. Let us, through an example, determine the practical method adopted in this type of problem situation.

Example 1

Knowing that a PA is formed by 20 numbers, where the₁ = 3 and the₂₀ = 79. Determine the arithmetic means existing between the₁ and the₂₀.

Let's determine the reason for this PA based on the following situation:

THE₂₀ = the₁ +19r
79 = 3 + 19r
79 – 3 = 19r
76 = 19r
r = 4

Knowing that the BP ratio is equal to 4, we will determine the numbers between the₁ and the₂₀:

3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79.

Example 2

In January, a company obtained a net profit equivalent to R$ 14,000.00. In that same year, in December, the net income was R$ 80,000.00. Knowing that the profit originated according to an increasing PA, determine the billing for the other months of the year.

January → The₁ = 14.000
December → The₁₂ = 80.000

THE₁₂ = the₁ + 11r
80,000 = 14,000 + 11r
80,000 - 14,000 = 11r
11r = 66000
r = 6000

Monthly breakdown of the company's billing:

January: BRL 14,000.00
February: BRL 20,000.00
March: BRL 26,000.00
April: BRL 32,000.00
May: BRL 38,000.00
June: BRL 44,000.00
July: BRL 50,000.00
August: BRL 56,000.00
September: BRL 62,000.00
October: BRL 68,000.00
November: BRL 74,000.00
December: BRL 80,000.00

by Mark Noah
Graduated in Mathematics
Brazil School Team

Progressions - Math - Brazil School

Source: Brazil School - https://brasilescola.uol.com.br/matematica/interpolacao-meios-aritmeticos.htm