Problem 31 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Sum to \(n\) terms of the series \(\frac{1}{5 !}+\frac{1 !}{6 !}+\frac{2 !}{7 !}+\frac{3 !}{8 !}+\ldots\) is(A) \(\frac{2}{5 !}-\frac{1}{(n+1) !}\) (B) \(\frac{1}{4}\left(\frac{1}{4 !}-\frac{n !}{(n+4) !}\right)\) (C) \(\frac{1}{4}\left(\frac{1}{3 !}-\frac{3 !}{(n+2) !}\right)\) (D) None of these

Step-by-Step Solution

Answer

The sum of the series to \( n \) terms is option (B): \( \frac{1}{4}\left(\frac{1}{4!}-\frac{n!}{(n+4)!}\right) \).

Step 1: Understanding the Pattern of the Series

The given series is \( \frac{1}{5!} + \frac{1!}{6!} + \frac{2!}{7!} + \frac{3!}{8!} + \ldots \). Here, each term in the series can be expressed as \( \frac{k!}{(k+5)!} \). Where \( k \) starts from 0 for the first term.

Step 2: Derive the General Formula

The general term in the series is \( \frac{k!}{(k+5)!} \), which can be simplified. Notice that:\[ \frac{k!}{(k+5)!} = \frac{1}{(k+5)(k+4)(k+3)(k+2)(k+1)} \]This can be simplified using partial fractions if needed, but let us see if we can match with one of the given formulas.

Step 3: Matching to the Given Options

To match to the options: look at option (B),\[ \frac{1}{4} \left( \frac{1}{4!} - \frac{n!}{(n+4)!} \right) = \frac{1}{4} \left( \frac{1}{24} - \frac{n!}{(n+4)(n+3)(n+2)(n+1)} \right) \]This expression simplifies in the pattern of the series with the initial term adjustment corresponding to the series formulation, factoring out \( \frac{1}{4} \) aligns our series structure. This matches the series representation.

Key Concepts

Summation

Summation in mathematics involves adding a sequence of numbers or terms and is symbolized by the sigma notation, \( \Sigma \). In the context of series, summation is applied to calculate the total of the terms over a defined range. In our exercise, the summation of the series needs to be mapped onto one of the given formulas.

• The series is represented as \( \sum_{k=0}^{n} \frac{k!}{(k+5)!} \).
• Our task is to find the total sum up to a specific number \( n \)th term.
• Each term in the series depends on \( k \), which starts from 0 and increases with each term.

Understanding summation is crucial in recognizing the pattern of the series, as the calculations involve identifying how each term's value adds to the total sum. The art of finding this total boils down to simplifying terms so that they match known summation formulas or expressions.

Factorial

The factorial, denoted by an exclamation mark following a number \( n \), is a fundamental mathematical operation. It represents the product of all positive integers up to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

• Factorials appear frequently in series and can significantly affect the growth rate of terms.
• In our series, both numerator and denominator include factorial terms, such as \( k! \) and \( (k+5)! \).
• The series progression involves terms like \( \frac{1}{5!}, \frac{1!}{6!}, \frac{2!}{7!}, \) which underscore the relationship defined in factorial terms.

Since factorials grow rapidly, they play a critical role in determining how quickly each term decreases as \( k \) increases. This makes the series converge towards a particular sum when taken to infinity. Factorials provide not only the mathematical rigor but also the necessary tools to simplify series using known patterns.

Series Pattern Recognition

In mathematics, recognizing patterns within a series is a crucial skill that aids in efficient problem-solving. A series pattern provides insights into its structure, helping determine an expression for the general term or simplify the summation.

• To deduce the general formula for a series, identifying a repetitive structure is key.
• In this exercise, each term takes the form \( \frac{k!}{(k+5)!} \).
• This structure exhibits a regular pattern influenced by factorial increases in the denominator.

Identifying patterns means we can often see parts of a series where terms cancel out or fit into a known formula, as seen in option \( (B) \). This method provides a shortcut compelling enough to solve series problems with reduced complexity. Recognizing how a series aligns with familiar formulas makes pattern recognition a powerful tool in much of mathematical problem-solving.