A particular geometric sequence has strictly decreasing terms. After the first term, each successive term is calculated by multiplying the previous term by m/7. If the first term of the sequence is positive, how many possible integer values are there for m?
decreasing terms means ... (m / 7) < 1
To find the number of possible integer values for m, we need to analyze the conditions under which the terms of the geometric sequence will be strictly decreasing.
Let's denote the first term as a₁. Given that the terms after the first term are calculated by multiplying the previous term by m/7, we can write the second term as a₂ = a₁ * (m/7), the third term as a₃ = a₂ * (m/7), and so on.
For the sequence to be strictly decreasing, we need each successive term to be smaller than the previous term. Mathematically, this can be represented as:
a₂ < a₁
a₃ < a₂
a₄ < a₃
...
Substituting the expressions for a₂, a₃, and so on, we get:
a₁ * (m/7) < a₁
a₁ * (m/7) * (m/7) < a₁ * (m/7)
(a₁ * (m/7))^2 < a₁ * (m/7)
Let's simplify this inequality:
(m/7)^2 < 1
Taking the square root of both sides, we get:
|m/7| < 1
Since the square root of a positive number is always positive, we can remove the absolute value signs:
m/7 < 1 and -(m/7) < 1
Simplifying these inequalities:
m < 7 and -m < 7
The first inequality, m < 7, implies that m can be any integer less than 7. So possible integer values for m are: -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, and 6.
However, we also need to consider the second inequality, -m < 7. Since -m is always less than or equal to 0, this inequality is always satisfied. Therefore, it does not impose any additional constraints on the value of m.
In conclusion, there are 13 possible integer values for m: -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, and 6.