Question

Find a formula for the following sum: 1/1.2 + 1/2.3 + ...+1/n(n+1) by examining the values of this expression for small values of n. You may want to find the sum for n1,2,3,... and then conjecture.

Answer 1

To find a formula for the sum: 1/1.2 + 1/2.3 + ...+ 1/n(n+1), we can examine the values of this expression for small values of n and then make a conjecture. The sum formula is n² for all positive integers n.

Step-by-step explanation:

To find a formula for the sum: 1/1.2 + 1/2.3 + ...+ 1/n(n+1), we can examine the values of this expression for small values of n and then make a conjecture. Let's calculate the sum for n = 1, 2, 3, and 4:

1. n = 1: The sum is 1/1.2 = 5/6.
2. n = 2: The sum is 1/1.2 + 1/2.3 = 5/6 + 5/21 = 25/21.
3. n = 3: The sum is 1/1.2 + 1/2.3 + 1/3.4 = 5/6 + 5/21 + 5/42 = 65/42.
4. n = 4: The sum is 1/1.2 + 1/2.3 + 1/3.4 + 1/4.5 = 5/6 + 5/21 + 5/42 + 5/60 = 205/60.

By examining these values, we can observe that the sum seems to be equal to n². To prove this, let's use mathematical induction:

Base case: For n = 1, the sum is 1/1.2 = 5/6 = 1².

Inductive step: Assume that the sum for n terms is equal to n². We need to show that the sum for (n + 1) terms is equal to (n + 1)². Adding 1/(n+1)(n+2) to the sum for n terms, we get:

n² + 1/(n+1)(n+2) = (n(n+1) + 1)/(n+1)(n+2) = (n² + n + 1)/(n+1)(n+2) = (n+1)².

Therefore, we can conclude that the sum: 1/1.2 + 1/2.3 + ...+ 1/n(n+1) = n² for all positive integers n.