Analyzing Mathematical Statements

The question asks us to determine which of the given mathematical statements is correct. We will analyze each statement individually.

Analyzing Statement I: Sum of a Series

Statement I gives the value of the expression \(100^2 - 99^2 + 98^2 - 97^2 + 96^2 - 95^2 + 94^2 - 93^2 + \dots + 22^2 - 21^2\).

This series consists of terms in the form \(a^2 - b^2\), where \(a\) and \(b\) are consecutive integers. We can use the difference of squares formula: \(a^2 - b^2 = (a-b)(a+b)\).

Let's apply this formula to each pair of terms:

• \(100^2 - 99^2 = (100 - 99)(100 + 99) = 1 \times 199 = 199\)
• \(98^2 - 97^2 = (98 - 97)(98 + 97) = 1 \times 195 = 195\)
• \(96^2 - 95^2 = (96 - 95)(96 + 95) = 1 \times 191 = 191\)
• ...
• \(22^2 - 21^2 = (22 - 21)(22 + 21) = 1 \times 43 = 43\)

The given expression is the sum of these results:

\(199 + 195 + 191 + \dots + 43\)

This is an arithmetic progression (AP). Let's identify its properties:

• First term (\(a_1\)): \(199\)
• Common difference (\(d\)): \(195 - 199 = -4\)
• Last term (\(a_n\)): \(43\)

To find the sum of this AP, we first need to determine the number of terms (\(n\)). The terms in the sum correspond to the pairs \((100, 99), (98, 97), \dots, (22, 21)\). The first number in each pair forms the sequence \(100, 98, 96, \dots, 22\). This is also an AP with first term 100, last term 22, and common difference -2.

Using the formula for the \(n\)-th term of an AP, \(a_n = a_1 + (n-1)d\):

\(22 = 100 + (n-1)(-2)\)

\(22 - 100 = -2(n-1)\)

\(-78 = -2(n-1)\)

\(39 = n-1\)

\(n = 40\)

So, there are 40 terms in the sum \(199 + 195 + 191 + \dots + 43\).

Now, we can calculate the sum of this AP using the formula \(S_n = \frac{n}{2}(a_1 + a_n)\):

\(S_{40} = \frac{40}{2}(199 + 43)\)

\(S_{40} = 20(242)\)

\(S_{40} = 4840\)

The calculated value matches the value given in Statement I. Therefore, Statement I is correct.

Analyzing Statement II: Simplifying an Algebraic Expression

Statement II gives the value of the expression \(\left(k^2+ \frac{1}{k^2} \right) \left(k- \frac{1}{k} \right) \left(k^4+ \frac{1}{k^4} \right) \left(k+ \frac{1}{k} \right)\left(k^4- \frac{1}{k^4} \right)\) as \(k^{16}- \frac{1}{k^{16}}\).

Let's simplify the given expression using the difference of squares formula \( (a-b)(a+b) = a^2 - b^2 \) and rearrange the terms strategically:

Expression = \(\left(k- \frac{1}{k} \right) \left(k+ \frac{1}{k} \right) \left(k^2+ \frac{1}{k^2} \right) \left(k^4+ \frac{1}{k^4} \right) \left(k^4- \frac{1}{k^4} \right)\)

First, combine the first two terms:

\(\left(k- \frac{1}{k} \right) \left(k+ \frac{1}{k} \right) = k^2 - \left(\frac{1}{k}\right)^2 = k^2 - \frac{1}{k^2}\)

Substitute this back into the expression:

Expression = \(\left(k^2 - \frac{1}{k^2} \right) \left(k^2+ \frac{1}{k^2} \right) \left(k^4+ \frac{1}{k^4} \right) \left(k^4- \frac{1}{k^4} \right)\)

Next, combine the first two factors (which came from the first three terms of the original rearranged expression):

\(\left(k^2 - \frac{1}{k^2} \right) \left(k^2+ \frac{1}{k^2} \right) = (k^2)^2 - \left(\frac{1}{k^2}\right)^2 = k^4 - \frac{1}{k^4}\)

Substitute this back into the expression:

Expression = \(\left(k^4 - \frac{1}{k^4} \right) \left(k^4+ \frac{1}{k^4} \right) \left(k^4- \frac{1}{k^4} \right)\)

Now, combine the first two factors:

\(\left(k^4 - \frac{1}{k^4} \right) \left(k^4+ \frac{1}{k^4} \right) = (k^4)^2 - \left(\frac{1}{k^4}\right)^2 = k^8 - \frac{1}{k^8}\)

Substitute this back into the expression:

Expression = \(\left(k^8 - \frac{1}{k^8} \right) \left(k^4- \frac{1}{k^4} \right)\)

This is the simplified form of the expression. It is not equal to \(k^{16} - \frac{1}{k^{16}}\). For the expression to simplify to \(k^{16} - \frac{1}{k^{16}}\), the factors would typically be of the form \(\left(k - \frac{1}{k}\right)\left(k + \frac{1}{k}\right)\left(k^2 + \frac{1}{k^2}\right)\left(k^4 + \frac{1}{k^4}\right)\left(k^8 + \frac{1}{k^8}\right)\).

Therefore, Statement II is incorrect.

Conclusion

Based on our analysis, Statement I is correct, and Statement II is incorrect.

The correct option is the one that states only Statement I is correct.

Revision Table: Key Concepts Used

Additional Information: Extending Algebraic Identities

The repeated application of the difference of squares formula is a powerful technique in simplifying certain types of algebraic products. Consider the product \(\left(x - y \right)\left(x + y \right)\left(x^2 + y^2 \right)\left(x^4 + y^4 \right)\dots\left(x^{2^m} + y^{2^m} \right)\).

This simplifies as follows:

• \(\left(x - y \right)\left(x + y \right) = x^2 - y^2\)
• \(\left(x^2 - y^2 \right)\left(x^2 + y^2 \right) = (x^2)^2 - (y^2)^2 = x^4 - y^4\)
• \(\left(x^4 - y^4 \right)\left(x^4 + y^4 \right) = (x^4)^2 - (y^4)^2 = x^8 - y^8\)

Continuing this pattern, the entire product simplifies to \(x^{2^{m+1}} - y^{2^{m+1}}\). In Statement II, if the expression had been \(\left(k - \frac{1}{k}\right)\left(k + \frac{1}{k}\right)\left(k^2 + \frac{1}{k^2}\right)\left(k^4 + \frac{1}{k^4}\right)\left(k^8 + \frac{1}{k^8}\right)\), it would simplify to \(k^{16} - \frac{1}{k^{16}}\). The presence of the \(\left(k^4- \frac{1}{k^4}\right)\) term as a separate factor changes the final simplified form.