How to find the difference in arithmetic progression

The concept of numerical sequence implies correspondence to each natural number of some valid value. Such a number of numbers can be both arbitrary and possess certain properties - progression. In the latter case, each subsequent element (member) of the sequence can be calculated using the previous one.

Arithmetic progression is a sequence of numerical values \u200b\u200bin which its neighboring members differ from each other to the same number (all elements of a series, starting from 2nd) possess the property. This number is the difference between the previous and subsequent member - constantly and is called the difference in progression.

1

Progression difference: Definition

Consider a sequence consisting of J values \u200b\u200bA \u003d A (1), A (2), A (3), A (4) ... A (J), J belongs to a set of natural numbers N. Arithmetic progression, according to its definition, - sequence , in which A (3) - A (2) \u003d A (4) - A (3) \u003d A (5) - A (4) \u003d ... \u003d A (j) - a (j-1) \u003d d. The value of D is the desired difference in this progression.

d \u003d a (j) - a (j-1).

Allocate:

• Increasing progression, in this case D\u003e 0. Example: 4, 8, 12, 16, 20, ...
• Descending progression, then D \u003c0. Example: 18, 13, 8, 3, -2, ...

2

The difference of progression and its arbitrary elements

If there are 2 arbitrary terms of progression (i-th, Kh), then set the difference for this sequence can be based on the relationship:

a (i) \u003d a (k) + (i - k) * d, it means D \u003d (A (I) - A (K)) / (I - K).

3

The difference of progression and its first term

How to calculate the desired progression difference (D) if its first element is known and an arbitrary other? Use the ratio A (k) \u003d A (1) + D (K - 1). Then d \u003d (A (k) - a (1)) / (k - 1).

This expression will help determine the unknown value only in cases where the number of the sequence element is known.

4

The difference of progression and its amount

The amount of progression is the sum of its members. To calculate the total value of its first J elements, use the corresponding formula:

S (j) \u003d ((A (1) + A (j)) / 2) * j, but because a (j) \u003d a (1) + d (j - 1), then S (j) \u003d ((A (1) + A (1) + D (J - 1)) / 2) * J \u003d (( 2a (1) + d (- 1)) / 2) * j.

Thus, to determine the difference D, it is possible to use the known value of the progression sum of S (J):

d \u003d ((s (j) - j * a (1)) / (j (j - 1))) * 2.