Discrete Structure Calculate The Following Sum 2 I

Calculating the sum of a discrete series with terms like 2i is a fundamental operation in mathematics and computer science. This guide explains how to compute such sums accurately, when they're needed, and how to interpret the results.

How to Calculate the Sum

To calculate the sum of a series where each term is 2i (where i represents the term number), you'll need to know the number of terms in the series. The sum can be calculated using the formula for the sum of an arithmetic series.

Key Formula

The sum S of the first n terms of a series where each term is 2i is given by:

S = 2(1 + 2 + 3 + ... + n) = 2 × (n(n+1)/2) = n(n+1)

This formula works because the series 1+2+3+...+n is the sum of the first n natural numbers, which has a known formula. Multiplying by 2 gives the sum of the series where each term is twice its position.

The Formula

The formula S = n(n+1) comes from the mathematical properties of arithmetic series. Here's how it's derived:

The sum of the first n natural numbers is n(n+1)/2.
Each term in our series is twice its position, so we multiply the sum by 2.
This gives us the final formula: S = 2 × (n(n+1)/2) = n(n+1).

This formula is valid only when the series starts with i=1. If the series starts with a different index, you'll need to adjust the formula accordingly.

Worked Example

Let's calculate the sum of the first 5 terms where each term is 2i.

Example Calculation

The series is: 2×1 + 2×2 + 2×3 + 2×4 + 2×5 = 2 + 4 + 6 + 8 + 10

Sum = 2 + 4 = 6; 6 + 6 = 12; 12 + 8 = 20; 20 + 10 = 30

Using the formula: S = 5 × (5+1) = 5 × 6 = 30

Both methods give the same result, confirming the formula's accuracy.

Interpreting Results

The sum calculated using this formula represents the total value of the series where each term is twice its position. This is useful in:

Computer science algorithms that process sequences
Mathematical modeling of discrete systems
Financial calculations involving weighted series

Understanding when to use this formula helps in solving more complex problems involving discrete series.

FAQ

What if the series doesn't start at i=1?: If the series starts at a different index, you'll need to adjust the formula to account for the starting point.
Can this formula be used for negative numbers?: Yes, the formula works for negative integers as well, following the same arithmetic series properties.
What if I need to calculate partial sums?: For partial sums, you can use the formula for the sum of the first n terms and subtract the sum of the first m terms.
Is this formula valid for non-integer values?: No, this formula is specifically for integer values of i. For non-integer values, you would need a different approach.