Question

Evaluate \(\Sigma 2^{i}\), where \(i=2,3,4 \ldots 10\). (1) 2044 (2) 2048 (3) 1024 (4) 1022

Short answer

Answer: (1) 2044

Step-by-step solution

Identify the geometric series

We can identify this as a geometric series because the ratio between consecutive terms is constant (in this case, the ratio is 2). The series can be written as: \(2^2 + 2^3 + 2^4 + \ldots + 2^{10}\).

Find the sum using the formula for the sum of a geometric series

The formula for the sum of a geometric series is given by: \(S_n = \frac{a_1 (1 - r^n)}{(1 - r)}\), where \(S_n\) is the sum of the series, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. In this case, we have: \(a_1 = 2^2 = 4\), \(r = 2\), and \(n = 10 - 2 + 1 = 9\). Now, we can plug these values into the formula to find the sum of the series: \(S_9 = \frac{4(1 - 2^9)}{(1 - 2)}\)

Simplify and find the correct answer

Now, let's simplify the expression: \(S_9 = \frac{4(1 - 2^9)}{(-1)} = \frac{4(-511)}{(-1)} = 2044\) So, the sum of the series \(\Sigma 2^{i}\), where \(i=2,3,4 \ldots 10\) is 2044. The correct answer is (1) 2044.

Key concepts

Mathematical Series

A mathematical series is a sum of terms that follow a particular mathematical rule or pattern. It's like a sequence, but instead of listing the members, we're adding them up. There are various types of series, each with its own distinctive properties, uses, and ways of finding the sum.

In the context of the exercise, the series in question is a geometric series, which is made up of terms that are each some constant multiple of the previous term. Understanding the structure of these series is crucial for students, as it opens doors to solving complex problems across different areas of mathematics and its applications in physics, economics, and beyond.

Exponential Sums

Exponential sums involve the addition of numbers where the exponent changes, usually following a consistent rule. Each term in the sum is an exponentiation where the base remains constant, but the exponent varies. For example, the sum of powers of 2 from 2² to 2¹⁰ is an exponential sum.

These sums have significant applications in computer science, electronics, and the analysis of algorithms, where dyadic (related to powers of 2) calculations frequently occur. Students will benefit from recognizing exponential growth patterns, as it enables them to estimate large sums quickly and understand the concept of exponential growth - a foundational concept for a variety of technological applications.

Sum of Powers of 2

The sum of powers of 2 is a common type of series, which we encounter both in pure mathematics and computer science. As the name suggests, each term is a power of 2. This series is especially important because binary systems, which are the basis of all modern computing, work based on powers of two.

In the exercise example, summing powers of 2 from 2² to 2¹⁰ may seem daunting at first glance, but the solution method demonstrates how geometric series formulas can simplify the process. Recognizing these patterns and understanding their sums is critical for students who wish to excel in fields involving binary logic and data storage.

Arithmetic Progression

An arithmetic progression is a series where the difference between consecutive terms is constant. This is unlike our earlier example; however, it's important to introduce for comparative purposes. Each term in the progression is composed by adding or subtracting the same fixed number to the preceding term.

Although the series in the exercise isn't an arithmetic progression, understanding its structure as distinct from geometric or exponential series will help students to quickly identify and apply the correct formula for evaluating sums. Arithmetic series have a wide range of applications in daily life problems, such as calculating the total interest over time, planning budgets, or even in architecture.