Question

Use induction to prove that, for n≥1 :
1/2+...+1/2ⁿ = 2ⁿ-1/ 2ⁿ
n=__

Answer 1

We can use mathematical induction to prove the given statement: 1/2 +...+ 1/2ⁿ = 2ⁿ-1/2ⁿ.

Step-by-step explanation:

We can use mathematical induction to prove the given statement:

Step 1: Base case - Let's check if the statement is true for n = 1.

Substituting n = 1 into the given expression, we get:

1/2 = 2^1-1/2^1 = 1/2

So, the base case holds.

Step 2: Inductive step - Assume that the statement is true for some positive integer k, i.e.,

1/2 + ... + 1/2^k = 2^k-1/2^k

Now, we need to prove that the statement is true for k+1:

1/2 + ... + 1/2^k + 1/2^(k+1) = 2^(k+1)-1/2^(k+1)

To prove this, we can rewrite the expression on the left-hand side as:

2^k-1/2^k + 1/2^(k+1)

Combining the fractions, we get:

(2^k-1 * 2)/(2^k * 2)

Simplifying, we have:

(2^(k+1)-2)/(2^(k+1))

Which is equal to:

2^(k+1)-1/2^(k+1)

Thus, the statement holds for k+1 as well.

Therefore, by the principle of mathematical induction, the statement holds for all positive integers n.