Arithmetic sequence question?

Question

For which values of x would the expression -8. x2 and 17x form the first three terms of an arithmetic series?

I absolutely do not understand the question

Answer 1

a1 = -8
a2 = x²
a3 = 17x

These are to be terms of an arithmetic sequence. That means that a3 - a2 = a2 - a1 = d, the common difference. So

a3 - a2 = a2 - a1
17x - x² = x² - (-8)
2x² - 17x + 8 = 0

Solve for x.

Answer 2

For an arithmetic sequence the ratio will be [x^2 - (-8)] or
17x - x^2

Then,

x^2 + 8 = 17x - x^2 (the ratio is always the same in this series)

Solving the equation
2x^2 - 17x + 8 = 0
2x^2 - 16x - x + 8 = 0
2x(x - 8) - (x - 8) = 0
(x - 8) (2x - 1) = 0

x1 = 8
x2 = 1/2

Both values (8 and 1/2 for x) form the first 3 terms of the series:

- 8, 64, 136
- 8, 1/4, 17/2

Answer 3

For any three consecutive terms in an arithmetic series, twice the middle term = the sum of the first term and the third term. Therefore, we have
2x^2 = -8+17x, x > 0
Collect all the terms in one side,
2x^2-17x+8 = 0
(2x-1)(x-8) = 0
x = 1/2, 8

Answer 4

ok so an arithmetic sequence runs along the lines of

a0, a0+k, a0 +2k, a0+3k ....

so we have a0 = -8 (first term that you have)

a1 = a0 + k = x^2 => x^2=k-8 => k=x^2+8

a2 = a0 + 2k = 17k => 17x = 2k-8 => 2k-17x-8=0

2(x^2+8) - 17x -8 = 0

2x^2 -17x + 8 = 0

2x - 1 = 0 =>x=1/2 => k = 8.25

or

x - 8 = 0 => x = 8 => k = 72

Now go back and check both the answers - see if they fit....

You will find that only the second arrangement works (why?) so that the arithmetic sequence is

-8, +64, +136. You can prove this for yourself.

hth