Question

Solve \(T(n)=T(n-1)+n\) for \(n \geq 1\) with \(T(0)=7\).

Short answer

The solution is \(T(n) = 7 + \frac{n(n+1)}{2}\).

Step-by-step solution

Identify the Relation Type

The given recurrence relation is linear and non-homogeneous, as it includes a non-recursive term, "+n". The base case provided is \(T(0) = 7\).

Establish the Recursive Steps

We need to compute the first few terms to observe any pattern. Start from the base case.- \(T(0) = 7\)- \(T(1) = T(0) + 1 = 7 + 1 = 8\)- \(T(2) = T(1) + 2 = 8 + 2 = 10\)- \(T(3) = T(2) + 3 = 10 + 3 = 13\)

Derive a General Formula

Observe the pattern from the calculations: \(T(n) = 7 + 1 + 2 + 3 + \ldots + n\). This sum is a well-known sequence, the sum of the first \(n\) natural numbers, given by the formula \(\frac{n(n+1)}{2}\).

Incorporate the Base Case

Combine the sequence formula with the initial term to generalize the solution: \[T(n) = 7 + \frac{n(n+1)}{2}\]

Verify the Solution

Verify by substituting initial values:- For \(n = 0\), \(T(0) = 7 + \frac{0(0+1)}{2} = 7\)- For \(n = 1\), \(T(1) = 7 + \frac{1(1+1)}{2} = 8\)- For \(n = 2\), \(T(2) = 7 + \frac{2(2+1)}{2} = 10\) The calculated values match those found in Step 2.

Key concepts

Initial Conditions

Initial conditions play a fundamental role in solving recurrence relations. They serve as the starting point for evaluating the recursive function. In the problem presented, the initial condition is given by \( T(0) = 7 \). This is essential because without this information, we couldn't accurately determine the values of \( T(n) \) for any \( n \geq 1 \).

Initial conditions are the building blocks of recursive sequences. They allow us to start the process of calculating subsequent terms. For instance, knowing \( T(0) = 7 \) enables us to find \( T(1), T(2), \) and so on, by following the recurrence \( T(n) = T(n-1) + n \).

• They provide a known anchor or reference point in the sequence.
• Help in uniquely determining the sequence for all \( n \) based on this predefined starting value.
• They set the stage for understanding how the recursive function operates over the domain.

In practice, always remember to check the initial condition when you're dealing with recurrence problems as it ensures the consistency and correctness of the solution.

Sum of Natural Numbers

The sum of the first \( n \) natural numbers is a classic problem in mathematics. It is frequently used in analyzing the structure of recurrence relations, just like in the current exercise. The sum of natural numbers is expressed by the formula \( \frac{n(n+1)}{2} \).

In our problem, the recurrence relation \( T(n) = T(n-1) + n \) necessitated the identification of this classic sequence. The pattern observed in the calculated terms eventually led us to realize that summing numbers from 1 up to \( n \) was equivalent to our non-recursive part of the formula.

This formula is efficient and can greatly simplify your calculations:

• Instead of adding each term individually, use the formula to quickly calculate the total sum.
• It's derived from the concept of pairing numbers efficiently, originally proposed by mathematicians like Gauss.
• Thus, the general solution for \( T(n) \) becomes \( 7 + \frac{n(n+1)}{2} \), incorporating both the initial condition and the sum of natural numbers.

Using this formula is a powerful tool in recognizing patterns within a sequence and solving recurrence relations that involve direct sums of natural numbers.

Recursive Functions

Recursive functions are a method of defining sequences or relations where subsequent values are defined based on preceding ones. In this scenario, \( T(n) = T(n-1) + n \) is a simple recursive function.

Understanding recursive functions involves:

• Identifying the base case, which in our example is \( T(0) = 7 \).
• Recognizing how each term depends on the previous term, which in this case is incremented by \( n \).
• Deriving a pattern or formula by unrolling or calculating a few terms, which helps in finding a closed-form solution.

Recursive functions typically offer a concise way to express a sequence where a small change or increment is applied consistently. Their power lies in their ability to describe potentially complex sequences with relatively simple rules.

In this case, the recurrence relation provides a clear structure with a repetitive and predictable process, lending itself to various applications in computer science and mathematics. Recursion alongside initial conditions helps uniquely define the progression of terms within the sequence.