Final answer:
To find the formula for the summation Σk=1n 1k(k+1), we observe that each term cancels out the next due to it being a telescoping series, resulting in the simplified form n/(n+1).
Step-by-step explanation:
The question asks to find a formula for the summation Σ₁⁺¹ ⅟ₖ(ₖ+1) by examining the value of this expression for small values of n. This summation can be further analyzed by recognizing each term as the difference of two fractions with a common denominator. For instance, ⅟ₖ(ₖ+1) simplifies to (ₖ+1)−ₖ divided by ₖ(ₖ+1). Doing this for several terms and adding them up shows each term cancels out the next, resulting in a telescoping series. Only the first term of the first fraction and the last term of the last fraction remain.
Thus, the series simplifies to 1 − ⅟(ₖ+1), which simplifies to n/(n+1). So the correct answer is option a (n/n+1).