Problem 110 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

In a G.P. if the \((m+n)\) th term be \(p\) and \((m-n)\) th term be \(q\), then prove that its \(m\) th term is \(\sqrt{p q}\).

Step-by-Step Solution

Answer

In a G.P., given that the \((m+n)\)th term is \(p\) and the \((m-n)\)th term is \(q\), the \(m\)th term, \(T_m\), is equal to \(\sqrt{p*q}\) under the condition that the common ratio \(r\) satisfies \(r^{2n}= 1\). This condition can be met if \(r=1\) or \(n = 0\).

Step 1: General term of a G.P.

The general term formula for a G.P. with the first term 'a' and common ratio 'r' is as follows: \[T_n = ar^{n-1}\] where \(T_n\) is the \(n\)th term of the G.P.

Step 2: Finding the \((m+n)\)th and \((m-n)\)th term

Using the general term formula, let's find the \((m+n)\)th and \((m-n)\)th terms of the G.P.: \(T_{m+n} = ar^{m+n-1}\) \(T_{m-n} = ar^{m-n-1}\) Now, substitute the values \(p\) and \(q\) for the \((m+n)\)th and \((m-n)\)th terms respectively: \(p = ar^{m+n-1}\) \(q = ar^{m-n-1}\)

Step 3: Finding the ratio of p and q

We will now divide the equation for \(p\) by the equation for \(q\) to find the ratio: \(\frac{p}{q} = \frac{ar^{m+n-1}}{ar^{m-n-1}}\) Simplifying the expression, we get: \(\frac{p}{q} = r^{2n}\)

Step 4: Finding the mth term formula

Now, let's find the \(m\)th term formula using the general term formula: \(T_m = ar^{m-1}\) We need to prove that \(T_m = \sqrt{p*q}\) Squaring both sides of the \(m\)th term formula: \((T_m)^2 = (ar^{m-1})^2\) Now, let's replace the \((ar^{m-1})^2\) with the expression we found in Step 3, which involves \(p\) and \(q\): \((T_m)^2 = pqr^{2n}\) Taking the square root of both sides, we get: \(T_m = \sqrt{p * q * r^{2n}}\)

Step 5: Conclusion

Since we have shown that the \(m\)th term of the G.P. can be written as \(T_m = \sqrt{p * q * r^{2n}}\), this implies that the \(m\)th term \(T_m\) is indeed equal to \(\sqrt{p*q}\) under the condition that \(r^{2n}= 1\). This can be satisfied if \(r=1\) or \(n = 0\). So, the given statement holds true under the specified conditions.

Key Concepts

Sequence and Series

Understanding sequences and series is crucial in mathematics, especially when tackling concepts such as geometric progressions (G.P.). Sequences are ordered lists of numbers following a defined pattern, while a series is the sum of the terms in a sequence. In G.P., each term is determined by multiplying the previous one by a fixed, non-zero number known as the 'common ratio'.

G.P. is part of a broader category that includes arithmetic progressions (A.P.), which add a fixed value to each term. Knowing the difference between these types helps in recognizing and solving various problems in mathematics. Both sequences and series find applications in real-world scenarios, from calculating loan repayments to predicting population growth.

General Term of G.P.

To solve problems involving G.P., it's essential to understand the formula for its general term. The formula is given by:
\[ T_n = ar^{n-1} \]
where \( T_n \) represents the \( n \)th term, \( a \) is the first term, and \( r \) is the common ratio. This formula allows a direct calculation of any term in the progression, once the first term and common ratio are known.

For example, in the problem given, the terms \( T_{m+n} \) and \( T_{m-n} \) are calculated using this same formula. Recognizing how to use this formula is vital for deducing and proving expressions involving terms of G.P. It's the key that opens up deeper exploration into the properties and behavior of geometric sequences.

Proofs in Algebra

Proofs in algebra often demand a step-by-step approach to establish the validity of equations or inequalities. They are crucial in demonstrating relationships and solving problems logically. In the given exercise, the proof shows that the \( m \)th term \( T_m \) of a G.P. is \( \sqrt{pq} \) under specific conditions.

To construct such proofs, begin by expressing known quantities using appropriate algebraic formulas. For instance, derive relationships between terms using their expressions as shown in the problem solution. Another common step is manipulation of these expressions through algebraic operations like division or multiplication, seeking either simplification or a necessary condition.

The proof concluded by relating the expressions with known constraints (like \( r^{2n} = 1 \)), highlighting the algebraic finesse required in these proofs. Proofs help in solidifying understanding and confirming that mathematical theories hold under set conditions, nurturing a deeper grasp of why and how solutions work.