What value of why I would make this a geometric sequence: y+ 1, y, y - 4....

Answer 1

A geometric sequence is characterized by (defined by, really) a common ratio between any 2 consecutive terms. In other words, any term in the sequence divided by the previous term should = the same number, a ratio we call r.

So, we will set up a proportion using this definition, then cross multiply to solve for y:

y / (y + 1) = (y - 4) / y

y² = (y + 1)(y - 4)

y² = y² - 3y - 4

3y = - 4

y = - 4/3

As a check: - 1/3 , - 4/3 , - 16/3. Yes, this forms a geometric sequence with t₁ = - 1/3 and r = 4.

Answer 2

A geometric sequence has the general form:

a_n = a₁·r^n-1

where a_n is the nth term in the sequence, a₁ is the first term in the sequence, and r is the common ratio. The common ratio can be found b y taking the ratio of any two successive terms:

r = a_n/a_n-1

You are given three terms in the sequence: y+1, y, and y-4. Thus we have:

y/(y+1) = (y-4)/y

Cross multiplying:

y² = (y-4)(y+1)

y² = y² - 3y - 4

Solve for y.