Answer:
Explanation:
To find the equation for the nth term in the sequence, we need to determine the pattern or rule that generates the sequence.
From the given information, we can see that the sequence is not arithmetic because the differences between consecutive terms are not constant. Instead, the sequence appears to be quadratic because the second difference between consecutive terms is constant.
Using the given values of a₁, a₃, and a₅, we can find the first few differences:
a₃ - a₁ = 4 - (-6) = 10
a₅ - a₃ = 14 - 4 = 10
So, the first difference is 10, which indicates a linear term in the equation for an. We can now use the given value of a₁ and the first difference to find the constant term in the quadratic equation. Let d be the common difference, then we have:
a₂ = a₁ + d = -6 + 10 = 4
a₄ = a₃ + d = 4 + 10 = 14
a₆ = a₅ + d = 14 + 10 = 24 - 1
a₇ = a₆ + d = 24 - 1 + 10 = 33
Now, we can find the second difference between consecutive terms:
a₄ - 2a₃ + a₂ = 14 - 2(4) + (-6) = 0
a₆ - 2a₅ + a₄ = 19 - 2(14) + 4 = -5
a₇ - 2a₆ + a₅ = 33 - 2(19) + 14 = 9
Since the second difference is constant (-5), this confirms that the sequence has a quadratic term in its equation. Let's assume the equation for the nth term is:
an = an² + bn + c
Substituting the values we know, we get three equations:
a₁ = a₁² + b₁ + c --> -6 = c
a₃ = a₃² + b₃ + c --> 4 = 9a + b + c
a₅ = a₅² + b₅ + c --> 14 = 25a + 5b + c
Solving this system of equations, we get:
a = 1/2, b = 19/2, and c = -6
Therefore, the equation for the nth term in the sequence is:
an = (1/2)n² + (19/2)n - 6.