Problem 279 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Sum the series \(1+2.2+3.2^{2}+4.2^{3}+\ldots+100.2^{99}\).

Step-by-Step Solution

Answer

The sum of the series is \(S = -100*2^{98} + 4\)

Step 1: Identify the progression

The given exercise represents an arithmetic-geometric series. In an arithmetic-geometric series, the difference between the consecutive arithmetic elements is constant and each term of the arithmetic series is multiplied by a constant ratio. Here, the arithmetic sequence is \(1,2,3,\ldots, 100\) and the geometric sequence is \(1, 2, 2^2,\ldots, 2^{99}\)

Step 2: Apply the formula

The sum \(S\) of such a series can be found using the formula: \(S = a\frac{1-r^n}{1-r}\) + \(\frac{d}{q}\)\((qr^n - nr^{n-1} - r)\), where \(a\) is the first term of the arithmetic series, \(d\) is the common difference, \(r\) is the common ratio, \(n\) is the number of terms, and \(q\) is the first term of the geometric sequence. In this case, \(a=1\), \(d=1\), \(r=2\), \(n=100\), and \(q=1\)

Step 3: Calculate

Substitute the values into the formula: \(S = 1*\frac{1-2^{100}}{1-2} + \frac{1}{1}(1*2^{100} - 100*2^{99} - 2)\), which simplifies to: \(S = -1*2^{100} + 2 + 2^{100} - 100*2^{99} + 2\), and further simplifies to: \(S = -100*2^{99} + 4\)

Step 4: Simplify

This can be simplified to the final answer: \(S = -100*2^{98} + 4\)

Key Concepts

Series Summation

In mathematics, a series is the sum of the terms of a sequence. Series summation involves adding up all the terms of a series to obtain a single value. When dealing with series, we often encounter different types that have specific rules and formulas.

• An arithmetic series sums the terms of an arithmetic progression, where each term is obtained by adding a constant difference to the previous term.
• A geometric series, on the other hand, sums the terms of a geometric progression, where each term is multiplied by a constant ratio to obtain the next term.

In the context of an arithmetic-geometric series, the terms are formed by multiplying the terms from an arithmetic progression with those from a geometric progression.
Summing such series, like the one from the provided exercise, often involves using specialized formulas that take into account both the arithmetic and geometric nature of the series. The final sum can be significantly simplified by recognizing these patterns.

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant to the previous term. This constant is known as the common difference, denoted by the letter \(d\). For example, the sequence \(1, 2, 3, \ldots, 100\) is an arithmetic progression where \(d = 1\).
Key properties of an arithmetic progression include:

• The first term is denoted by \(a\).
• The \(n^{th}\) term can be calculated using the formula \(a + (n-1)d\).
• The sum of the first \(n\) terms can be calculated using the formula \(\frac{n}{2}(2a + (n-1)d)\).

Understanding these properties is crucial for tackling problems that involve summing up terms in an arithmetic progression or when they form part of another type of series, like the arithmetic-geometric series in our exercise.

Geometric Progression

A geometric progression (GP) is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number, known as the common ratio, \(r\). For instance, the sequence \(1, 2, 4, 8, \ldots\) has a common ratio of 2.
The features of a geometric progression include:

• The first term is \(q\).
• The \(n^{th}\) term can be expressed as \(qr^{n-1}\).
• The sum of the first \(n\) terms of a geometric progression is given by the formula \(\frac{q(1-r^n)}{1-r}\) if \(r eq 1\).

Geometric progression plays a significant role in arithmetic-geometric series, where terms from a GP are combined with those from an AP. Recognizing and using the properties of GPs enables one to solve complex series-related problems, such as the one in our original exercise, with greater ease and accuracy.