Determine the value of x that makes this sequence arithmetic: x-4, 6, x.

The value of x is 8.

Expert Answers

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A sequence of numbers `a _ 1 , ` `a _ 2 , ` `a _ 3 , . . . ` is called arithmetic sequence if the difference between any adjacent terms is the same:

`a _ 2 - a_ 1 = a _ 3 - a _ 2 = a _ 4 - a _ 3 ` and so on.

This difference is called the common difference of the arithmetic sequence. It can be positive, negative, or even zero: the sequence `1 , 1 , 1 , 1 , . . . ` is also an arithmetic one. The terms of the sequence can also be any real numbers.

Now look at the given sequence: `x - 4 , ` `6 , ` `x . ` Actually, only three terms of the sequence is given. But if we know that it is an arithmetic one, it is sufficient to find `x ` and to predict all other terms.

Indeed, three terms have two differences between adjacent terms, `6 - ( x - 4 ) ` and `x - 6 . ` These differences must be equal, which gives us an equation for `x : `

`6 - ( x - 4 ) = x - 6 .`

It is easy to solve it: `6 - x + 4 = x - 6 , ` `10 - x = x - 6 , ` `2 x = 16 , ` `x = 8 .`

This way, the given part of the sequence becomes `4 , 6 , 8 , ` while the common difference itself is `2 . ` The next terms are next even integers.

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