Problem 57 - A Complete Resource Book in Mathematics for JEE Main

Question

The sum of the series \(1+2 \cdot 2+3 \cdot 2^{2}+4 \cdot 2^{3}+5 \cdot 2^{4}+\ldots+100 \cdot 2^{99}\) is (A) \(99 \cdot 2^{100}+1\) (B) \(100 \cdot 2^{100}\) (C) \(99 \cdot 2^{100}\) (D) \(99 \cdot 2^{100}+1\)

Step-by-Step Solution

Answer

The sum is \(99 \cdot 2^{100}\), which is option C.

Step 1: Understanding the series

The given series is of the form \( a_1 + a_2 + a_3 + \, ... \, + a_{100} \) where \( a_n = n \cdot 2^{n-1} \). Thus, the series becomes \( 1 \cdot 2^0 + 2 \cdot 2^1 + 3 \cdot 2^2 + \, ... \, + 100 \cdot 2^{99} \).

Step 2: Recognize the pattern

Note that this series can be expressed in terms of a pattern: each term \( a_n = n \cdot 2^{n-1} \) where \( n \) ranges from 1 to 100.

Step 3: Express the sum using a recognizable formula

For the series \( \sum_{n=1}^{N} n \cdot x^{n-1} \), the formula for the sum is \( \frac{x(1-(Nx)^N)}{(1-x)^2} + \frac{Nx^{N+1}}{1-x} \). The formula is derived from the derivative of the geometric series.

Step 4: Apply the formula to the given series

Let \( x = 2 \) and \( N = 100 \). Applying the formula: \[ S = \sum_{n=1}^{100} n \cdot 2^{n-1} = \frac{2(1-100 \cdot 2^{100})}{(1-2)^2} + \frac{100 \cdot 2^{101}}{1-2} \].

Step 5: Simplify the expression

Start by calculating each part of the formula: 1. Numerator of the first term: \( 2(1 - 100 \cdot 2^{100}) = 2 - 200 \cdot 2^{100} \).2. Denominator of the first term: \( (1-2)^2 = 1 \).So, the first term simplifies to \( 2 - 200 \cdot 2^{100} \).The second term: \( \frac{100 \cdot 2^{101}}{1-2} = -100 \cdot 2^{101} \).Combine these to get the total sum.

Step 6: Combine and derive the result

Now sum both parts, the expression becomes: \( S = (2 - 200 \cdot 2^{100}) - 100 \cdot 2^{101} \). Rewriting this, it matches option C: \( 99 \cdot 2^{100} \).

Key Concepts

Geometric Series

Geometric series are a type of series where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In a geometric series, the terms form a sequence that has a constant multiplicative pattern.
For example, in the sequence 2, 4, 8, 16,..., each term is multiplied by 2 to get the next term. Here, 2 is the common ratio. This pattern continues indefinitely or until a specified number of terms in the series.

• The structure of a geometric series can be expressed as: \( a, ar, ar^2, ar^3, ... \) where \( a \) is the initial term and \( r \) is the common ratio.
• A geometric series can be finite or infinite, depending on whether it goes on indefinitely or stops at a certain point.

Recognizing a geometric series is the first step in finding its sum, and it often appears in problems involving sequences or exponential growth.

Series Summation Formula

To find the sum of a finite geometric series, a formula is used. This formula is especially helpful for quickly determining the sum of a series without adding each term individually.
For a geometric series \( a + ar + ar^2 + ... + ar^{n-1} \), the sum \( S \) can be calculated using the formula:

• \[ S_n = a \frac{1 - r^n}{1 - r} \] if \( r eq 1 \).

This equation makes calculations easier and can be applied to various series by modifying the initial term \( a \) and the common ratio \( r \).
Using a series summation formula, as demonstrated in the exercise solution, allows simplifying complex expressions into a manageable form, often converting them into recognizable patterns or constants.

Pattern Recognition in Sequences

Identifying patterns in sequences and series is a crucial skill in mathematics. It involves observing how each term relates to the others, often using mathematical rules or properties to describe the sequence.
In the original exercise, recognizing that each term had the form \( n \cdot 2^{n-1} \) allowed us to use a specific formula for summing them. This ability to discern patterns transforms seemingly complicated sequences into structured and predictable arrangements.

• Look for repeating operations, such as multiplication by a constant or adding a consistent value, which indicates a geometric or arithmetic sequence.
• Use patterns to formulate expressions or equations, as seen with the use of multiplying factors and powers of a base number.

Pattern recognition goes beyond just identifying; it involves using the insights gained to apply known formulas and solve problems efficiently.