Problem 120 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the sum of the first ten terms of the series \(\left(1 \frac{3}{5}\right)^{2}+\left(2 \frac{2}{5}\right)^{2}+\left(3 \frac{1}{5}\right)^{2}+4^{2}+\left(4 \frac{4}{5}\right)^{2}+\ldots \ldots\), is \(\frac{16}{5} m\) then \(\mathrm{m}\) is equal to : [2016] (a) 100 (b) 99 (c) 102 (d) 101

Step-by-Step Solution

Answer

m = 101 (d)

Step 1: Identify the terms

First, write down the expression of each term in the series. Notice that each expression is of the form \((n + \frac{5-n}{5})^2\), with \(n\) being the term index. Therefore, it simplifies to \((n + \frac{5}{5} - \frac{n}{5})^2 = (n + 1 - \frac{n}{5})^2\).

Step 2: Simplify the terms

Simplify the general expression for the terms.\[((n + 1 - \frac{n}{5})^2) = ((\frac{5n+5-n}{5})^2)\]Simplifying further:\[((\frac{4n+5}{5})^2)\]

Step 3: Calculate sum of first ten terms

To find the sum of the first ten terms, plug \(n = 1, 2, ... , 10\) into the formula and calculate each term:- Term 1: \((\frac{9}{5})^2 = \frac{81}{25}\)- Term 2: \((\frac{13}{5})^2 = \frac{169}{25}\)- Continue this for terms 3 to 10.Add up these results to find the total sum \(S\).

Step 4: Use the given sum condition

According to the problem statement, the sum of these terms is \(\frac{16}{5}m\). Substitute \(S\) from Step 3 into this equation:\[S = \frac{16}{5}m\]

Step 5: Solve for m

Equate the calculated sum \(S\) to \(\frac{16}{5}m\) and solve for \(m\). Rearrange to find:\[m = \frac{5S}{16}\]Substitute \(S\) to find \(m\).

Step 6: Verify the numerical value

Ensure you've evaluated the numerical calculations correctly by matching the sum \(S\) with the solution of the equation for \(m\). This is done by back-calculating any steps if necessary to confirm correctness.

Key Concepts

Sum of Series

Series are collections of numbers, often following a pattern, that are summed together. Understanding how to find the sum of a series can be crucial for solving many mathematical problems.
For the series given in the problem, each term has a specific formula that allows us to compute its value. The steps to find the sum of a series can be streamlined as follows:

• Identify the pattern or rule for the terms of the series. This is often given explicitly or can be deduced.
• Calculate each term as per this rule for the number of terms required. In our exercise, this involves plugging in values from 1 to 10.
• Add the results of these calculations to find the total sum.

The challenge is not just in computing each term, but ensuring each step correctly contributes toward the desired total sum. This often involves simplifying expressions, as seen in our problem, to make calculations manageable.

Term Calculation

Calculating terms of a series can be straightforward or complex, depending on the rule governing the series. For the given exercise, each term is expressed using a formula:

• Our series uses the expression \((n + \frac{5}{5} - \frac{n}{5})^2\), which simplifies to \((\frac{4n+5}{5})^2\).
• This expression changes as ‘n’, the term index, varies from 1 to 10.
• The term calculation requires substituting different values of ‘n’ into this expression to find the corresponding term's value.

For example, for n = 1, the calculation yields \(\left(\frac{9}{5}\right)^2\), while for n = 2, it becomes \(\left(\frac{13}{5}\right)^2\). Each term calculation involves straightforward substitution and basic arithmetic operations such as squaring and division.
Understanding each step thoroughly is key, as any errors in substitution or arithmetic can lead to incorrect results quickly.

Quadratic Expressions

Quadratic expressions are mathematical expressions involving variables raised to the second power, or squared. These are particularly interesting in series where each term may involve quadratics.
The given series provides an excellent example. Each element of the series follows the pattern \(\left(\frac{4n+5}{5}\right)^2\). Let's break it down:

• The expression inside the square, \(\frac{4n+5}{5}\), involves both multiplication and division, making it necessary to approach simplification systematically.
• Squaring the expression requires applying the formula \(a^2 = b^2 + 2ab + c^2\) if written as an expanded form; however, here it's merely used directly as \(\left(\frac{4n+5}{5}\right)^2\).

This quadratic nature introduces an added layer of complexity due to the requirement to square fractions accurately which in turn affects precision in the sum calculation.
Paying attention to these details in manipulating quadratic expressions inside a series is crucial to arriving at the correct sum of the series.