Find the value of x so that the given terms are consecutive terms of an arithmetic sequence. `2x` , `3x+1` , and `x^2+2`
In arithmetic sequence, two consecutive terms has a common difference (d).
If the arithmetic sequence is 2x, 3x+1, x^2+2 , the common difference is:
`d= a_2-a_1 = 3x + 1 - 2x`
`d=x + 1`
`d = a_3-a_2=x^2+2-(3x+1)`
To solve for x, set the two d's equal to each other.
Express the equation in quadratic form `ax^2+bx+c=0` .
Factor left side.
Set each factor equal to zero. And isolate x.
`x-4=0 ` and `x=0`
Hence, there are two values of x that would give us an arithemtic sequence for the terms `2x` , `3x+1` and `x^2+2` . These are x=0 and x=4.