Problem 18 - Calculus

Question

Evaluate \(\sum_{i=1}^{n}\left(3^{i}-3^{i-1}\right)\).

Step-by-Step Solution

Answer

The sum evaluates to \(3^n - 1\).

Step 1: Expand the Expression

Begin by expanding the expression inside the sum: \[\sum_{i=1}^{n} (3^{i} - 3^{i-1}) = (3^1 - 3^0) + (3^2 - 3^1) + (3^3 - 3^2) + \ldots + (3^n - 3^{n-1}).\]

Step 2: Observe the Pattern

Notice that this sequence is telescoping. Most terms will cancel with each other when expanded. Each \(3^i\) term is subtracted in the next term, leading to extensive cancellation.

Step 3: Cancellation of Terms

Write out the expanded sum to better see the cancellation: \[(3^1 - 3^0) + (3^2 - 3^1) + (3^3 - 3^2) + \ldots + (3^n - 3^{n-1}).\]After cancellation, most terms vanish, leaving only \(-3^0 + 3^n\) from the beginning and end of the series.

Step 4: Simplified Result

After canceling as outlined in the previous step, we are left with: \[3^n - 1.\]

Step 5: Conclusion

The final value of the given sum is obtained from the simplified result in Step 4, which is \[3^n - 1.\]

Key Concepts

Mathematical Induction

Mathematical induction is an important concept in proving mathematical statements, particularly those involving sequences or series.
It provides a method to demonstrate that a given statement is true for all natural numbers.Here's a step-by-step look at how it generally works:

• **Base Case**: Start by proving that the statement is true for the first natural number, usually 1. This establishes the base.
• **Inductive Step**: Next, assume that the statement is true for some arbitrary natural number, say \(k\). This assumption is known as the induction hypothesis.
• **Proof for \(k+1\)**: Using the induction hypothesis, demonstrate that the statement is also true for \(k+1\). If you can show this, then by induction, the original statement is true for all natural numbers.

This method is potent because it simplifies proving complex series or sequence-related problems, just like telescoping series. In our exercise, although we did not explicitly use induction, it remains a vital concept for verifying series behavior for any natural number \(n\).

Series Expansion

Expanding a series involves writing it out in an understandable form, allowing for easier manipulation and analysis.
It is especially helpful in identifying patterns such as telescoping series, where many terms cancel out.When you expand the series \(\sum_{i=1}^{n}\left(3^{i}-3^{i-1}\right)\), you break it down into its smaller parts:\[(3^1 - 3^0) + (3^2 - 3^1) + (3^3 - 3^2) + \ldots + (3^n - 3^{n-1}).\]When looking at each term created by expansion, you'll notice that each positive term will eventually cancel with a subsequent negative term.
This results in most terms disappearing except for the first negative term \(-3^0\) and the last positive term \(3^n\).By expanding series in this manner, you gain deeper insight into its structure, making it easier to simplify and evaluate.

Expression Simplification

Simplifying expressions is crucial to solving series problems effectively.
It involves reducing expressions to their simplest form to quickly find the solution.In our telescoping series, the sequence \(\sum_{i=1}^{n} (3^{i} - 3^{i-1})\) simplifies dramatically.By writing out the series, you see significant cancellation.

• The sequence shows that every positive term \(3^i\) cancels with a similar negative term \(-3^{i-1}\) from the next term.
• After the cancellation, you are left with a much-simplified expression: \(3^n - 1\).

This streamlined form arises because only the initial negative element \(-3^0\) (i.e., \(-1\)) and the last positive element \(3^n\) remain.Notice how simplification through telescoping effectively reduces the complexity of the calculation.This technique is instrumental in solving series problems where terms naturally align to cancel each other.