# Terms of an arithmetic seriesDetermine x if x+1, x-1, 4 are terms of an arithmetic series.

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### 2 Answers

You may use the relation between consecutive terms of arithmetic progression, such that:

`a_n = a_(n-1) + d`

d represents the common difference

Reasoning by analogy, yields:

`{(x - 1 = x + 1 + d),(4 = x - 1 + d):}` => `{(x + d = x - 2),(5 = x + d):} => x - 2 = 5 => x = 7`

**Hence, evaluating x, under the given conditions, yields `x = 7` .**

To determine the terms of the given arithmetical progression, we'll have to find out the value of x, first.

The terms (x+1), (x-1) and 4 are the consecutive terms of the arithmetical progression if and only if the middle term is the arithmetical mean of the neighbor terms:

x - 1 = [(x+1) + 4]/2

We'll multiply by 2 both sides:

2x - 2 = x + 1 + 4

We'll combine like terms from the right side:

2x - 2 = x + 5

We'll subtract x+5:

2x - 2 - x - 5 = 0

x - 7 = 0

We'll add 7 both sides:

**x = 7**

**The terms of the arithmetical sequence, whose common difference is d = -2 are: x + 1 = 7+1 = 8 ; x- 1 = 7-1 = 6 ; 4.**