Question

Compute the values of the product (1+1/+ 1 + 1) --- (1+) for small values of n in order to conjecture a general formula for the product. Fill in the blank with your conjecture. (1 + -) 1 + X 1 + $) -

Answer 1

The general formula for the product (1+1/n)(1+1/n-1)...(1+1/3)(1+1/2)(1+1) for small values of n is n+1.

Step-by-step explanation:

The given expression is (1+1/n)(1+1/n-1)...(1+1/3)(1+1/2)(1+1).

To compute the product for small values of n, we can plug in values such as n=1, n=2, n=3, etc. and calculate the product.

For example, when n=1, the product is (1+1/1) = 2.

When n=2, the product is (1+1/2)(1+1/1) = (3/2)(2) = 3.

By continuing this process for different values of n, we can conjecture that the general formula for the product is n+1.

Answer 2

The question is about deducing a general formula for a product using series expansions and the binomial theorem. The steps involve systematically simplifying the series by adding and subtracting values from its terms. The resulting conjecture is that the expression simplifies to 2n^2.

Step-by-step explanation:

The question involves using a series expansion technique to find a general formula for a given mathematical expression. Based on the clues given, it seems like we should consider the expression as some form of a series that simplifies to a function of n. The use of the binomial theorem suggests that the series can be expanded and simplified step by step, using the coefficients from the expansion.

Through the series of steps given, we can deduce that each term in the series contributes to the final result of n2. By adding and subtracting specific values from the terms in the series, we obtain a simplified series where every term is simply n, which when summed yields the final answer of 2n2.