Step 1: Understand the Problem

The problem asks us to find the value of 'n' in the binomial expansion of (2x-1÷x)^n when the constant term is -160. To solve this problem, we need to understand the concept of binomial expansion and how to find the constant term.

Step 2: Understand Binomial Expansion

The binomial expansion is a way to expand a binomial expression raised to a power 'n'. It is given by the formula:

(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n) * a^0 * b^n

where C(n, r) represents the binomial coefficient, which is calculated as C(n, r) = n! / (r! * (n-r)!).

Step 3: Finding the Constant Term

The constant term in the binomial expansion corresponds to the term where the exponent of 'a' is 0. In our given expression, (2x-1÷x)^n, we can rewrite it as (2x/x - 1/x)^n = ((2x - 1) / x)^n.

To find the constant term, we need to set the exponent of 'a' to 0, which means setting n - 0 = 0. So, 'n' should be 0 to have a constant term.

Step 4: Solve for 'n'

Since the constant term in the expansion is -160, and the constant term appears when the exponent of 'a' is 0, we can substitute 'n' with 0 in the binomial expansion formula and solve for the constant term.

(2x)^0 * (-1)^n * C(0, n) = -160

Simplifying further, we get:

(-1)^n * C(0, n) = -160

Since C(0, n) is 1 for any value of 'n', we can simplify the equation to:

(-1)^n = -160

Step 5: Solve the Equation

To solve the equation (-1)^n = -160, we need to find a value of 'n' that makes the equation true. The exponent of -1 alternates between positive and negative values, so we can conclude that 'n' should be an odd number.

If we substitute n = 1 into the equation, we get (-1)^1 = -1, which is not equal to -160. Similarly, for n = 3, we get (-1)^3 = -1, which is also not equal to -160.

Hence, there is no value of 'n' that satisfies the equation (-1)^n = -160. Therefore, there is no solution to this problem.

Conclusion

In conclusion, after analyzing the given binomial expression and understanding the concept of binomial expansion, we found that there is no value of 'n' that satisfies the equation (-1)^n = -160. As a result, the problem does not have a solution.