find the nth term of the linear sequence (5x+1),(2x+1),(1-x)
To find the nth term of a linear sequence, we first need to identify the pattern in the sequence.
Looking at the given sequence (5x+1), (2x+1), (1-x), we can observe that each term is obtained by changing the coefficient of x and adding or subtracting a constant term.
From the first term to the second term, the coefficient of x is decreasing by 3 and the constant term remains the same.
From the second term to the third term, the coefficient of x is decreasing by 1 and the constant term is changing the sign.
Therefore, we can deduce that the pattern in the sequence is:
Term 1: (5x + 1)
Term 2: (2x + 1)
Term 3: (1 - x)
To find the nth term, we need to determine how the coefficients and constants change as we go from one term to the next.
First, let's examine the change in the coefficient of x. The initial term has a coefficient of 5, the second term has a coefficient of 2, and the third term has a coefficient of 1. It is clear that the coefficient is decreasing by 3 each time.
For the change in the constant term, we observe that it remains the same between the first and second terms (both are 1), and the sign changes between the second and third terms (from +1 to -1).
Given the pattern we have discovered, we can now write the nth term:
Coefficient of x = 5 - 3(n-1) = 5 - 3n + 3 = -3n + 8
Constant term = 1 if n is odd, and -1 if n is even
Thus, the nth term of the given linear sequence is (-3n + 8)x + (-1)^n.